I'm reading through Pomeranz's book right now and yes, his two major focus is fossil fuel (the availability of large amounts of coal at strategic places in Britain) and overseas expansion/colonization. Now the issue of Great Divergence is one that has been extensively studied by the California School, of which Pomeranz is a member of. Bin Wong's first book China Transformed looks at state-building and population in China and Europe and his second book Before and Beyond the Divergence focuses on political differences and institutional changes. Peter Perdue, who I just found out today is also a member of the California School, has also written about state-building in his book China Marches West, that touches a little bit on why China's policies led to disaster in the Opium War. If you're interested in reading about the Great Divergence, then I suggest you read these other literature as well. All of them agree on the basic premise, but how they get there is different.

Now going to your question about whether or not institutions played a role in the Great Divergence. I would argue that yes, they played a very big role. The Naito Hypothesis of the Song as an Early Modern state has become a bit controversial nowadays, but from an economic and fiscal point of view, changes in the Song was truly revolutionary compared to the Tang before it and the Yuan after. I suggest you read Liu Guanglin's The Making of a Fiscal State in Song China, 960-1279, as it does a very good job explaining the successes and failures of Song fiscal policies. Economic data from the Song is sparse, and we only have a handful of complete sets of economic data from the 1000s, but from what these data tell us, the Song drew in enormous amounts of revenues through indirect taxation and later through the issuing of proto-credit instruments, which later European states such as Britain took advantage of. One of the reasons why these economic policies were successful, as pointed out by Liu, was that they were backed by effective fiscal institutions.

To quote Liu:

In a land empire as large as Song China, the operation of such an administrative network required the development of professional tax services, including, to name but a few, the recruitment of a large number of administrative staff, the application of uniform tax rates, the enforcement of discipline by combining rewards with punishments, the establishment of tax quotas for local institutions, and the adoption of tax receipts to track goods shipped over long-distance routes. All of these measurements require sophisticated planning and complicated execution, yet the Song court put them into practice for over three centuries. This success supports Hartwell’s argument that only professional financial services could have transformed the traditional Chinese bureaucracy. In other words, a government could rule with money only when it became an effective instrument for taxing resources and launching public projects.

The State Finance Commission was established as a single authority that exercised overall responsibility for receiving and accounting for state revenues. In 978, it was reported to have comprised 24 offices and employed over 1,000 clerks. The commissioners (sansishi) at the head of the State Finance Commission were carefully chosen by the court. These commissioners reported annual revenues and expenditure to the court in a short statement, and a government budget known as kuaijilu was prepared as a result. To avoid corruption and inefficiency, the State Finance Commission was also responsible for auditing local financial administrations. Su Zhe (1039–112), the famous essayist and a high- ranking official in the central government, criticized the oversized body of this financial administration, claiming that ‘in the vast expanse of all the state’s territory, every hair’s worth of expenses must go through the State Finance Commission’. Eventually, as the centralization of the entire financial system developed, the problem of how to supervise the State Finance Commission became a crisis in its own right. In the end, a special institute within the commission known as gouyuan was established for internal auditing. However, problems remained.

The development of professional services also refers to the prevalence of the revenue centered meritocracy in the Song bureaucracy. Officials were appointed, reviewed, and rewarded based on their financial administration records.The chance of promotion in someone’s official career heavily depended on how much he could increase the share of indirect tax revenues contributed to the court, a key step in centralizing the financial administration. To ensure professionalism, the appointment of high officials in the central government had to demonstrate their financial expertise background. For most of the first century of the Northern Song dynasty (960–1127), over 75 per cent of the Council of the State members, the top level of the central authority, had previous experience in financial administration. Aside from the predictable pattern of routine promotions based on specialty and work experience, these professional elites were also expected to have ‘administrative ability, mathematical skill, a grasp of the principles of classical monetary analysis, and an understanding of subsequent refinements in Chinese economic thought’. Finance-related questions even found their way into the civil examinations to test the professional knowledge of candidates.

So we see from the Song that its economic success was in fact very much tied to its effective political institutions. Now why this economic success didn't lead to an Industrial Revolution is itself a very interesting topic, but one that is beyond the scope of your question. Because the Song needed to compete with rival states that was equal to itself militarily, it needed to maintain a large standing army. Unlike the Han and Tang, the Song army was a full mercenary army, which added huge expenses to the state. Because of this, the state needed to be very creative in order to gain revenue and revenue subsequently became very important to Song officials.

But the subsequent Yuan diverged from the Song model. The Yuan no longer gained its revenue primarily through trade, but rather through the land tax and the salt monopoly. It also brought back corvee labor as a form of taxation (corvee labor was abolished in the Song), which meant that the state had access to a huge pool of labor that added very little cost to the state. The military was also organized into hereditary households that were in theory supposed to be self-sufficient and self-replicating. Then there was the fact that the Mongol rulers lacked the administrative expertise required to operate the complex Chinese bureaucracy, leading to a decline in institutional quality. The Ming was even worse - the first emperor Zhu Yuanzhang severely retarded economic growth with his policies of autarky. Just as in the Yuan, the military was supposed to be entirely self-sufficient and self-replicating (a lofty goal to be sure, but one that was not realistic even in Zhu Yuanzhang's time) and corvee labor utilized for massive state projects. Therefore, its bureaucracy was also not one that was geared towards revenue collection or finding ways of raising revenue.

For the Qing, Peter Perdue made the argument that Qing institutions were geared towards expansion, but into Central Asia. Unlike the Song, the Qing never faced an enemy that was militarily equal or superior to itself, therefore it was never put into a position where it needed to compete constantly with rival states and develop the need for revenue. We must remember that the war with the Zunghars was fought for such a long time not because the Zunghars proved to be formidable foes, but rather because the Qing faced repeated supply constraints, which the Qing state attempted to address by introducing new institutions. But in any case, the Zunghar wars was all but over by the 1770s, and the Qing no longer had an external foe to deal with. According to Perdue, the Qing emulated its successful policies against the Zunghars in the southeastern coast against the British as well, and it ultimately failed because the British proved to be completely different foes and the situation was completely different.

Pomeranz in his book does also talk about the fact that the Qing never developed any public debt. Britain certainly developed the financial institutions for debt, in fact the American Revolution was triggered by taxes the British government levied in an attempt to offset the debt it had incurred during the Seven Years War. Traditional Chinese thought operates under the assumption that wars can only be fought if the treasury can support it, that is why many officials always complain about going to war since wars were expensive. The Ming ultimately collapsed because its revenues could not keep up with its need to make war - the Ming never developed complex financial institutions to draw in revenue. Its revenues came primarily from the land tax, which over time decreased as the amount of taxable land decreased. The High Qing regularly ran budget surpluses, so for most of that period expenditures was not a major problem. Domestic trade had been growing since the late Ming, but neither the Ming nor the Qing (until the 1860s) taxed domestic commerce. The Song, on the other hand, imposed excise and customs duties on consumption goods and that accounted for the vast majority of government revenues. The Southern Song also made very effective use of a proto-public credit instrument to raise revenue, but ultimately mismanagement caused the entire system to implode and led to hyperinflation.