This isn't an area of my speciality, but early modern financing comes up a lot in what I am interested in, so I'll throw down what thoughts I have.

Ultimately, there was nothing they could do if you were just a banker, whose loan to a sovereign had gone bad. As you say, you can't exactly go and repo a sovereign's possession, so if the sovereign defaults on their payments, you are just out of luck. However, a sovereign who defaults is going to find it much more difficult to borrow money in the future. Multiple states in the early modern went through these kind of 'credit crunches', where they were unable to secure sources to borrow money from.

In this respect, the sovereign's fear of bankruptcy leading to future difficulties in securing capital is going to be the banker's great tool. Simultaneously, careful planning of who you loan money to was necessary. If a particular sovereign had previously defaulted, then loaning money to them again might not be the wisest decision, or you are going to demand a very high interest rate to try and protect yourself.

I'm not aware of any court proceedings that would have been launched to try and compel payment, or any means by which bankers could try and enforce payment from a sovereign who had decided not to. However, when you are a state that is dependent on the financial system to maintain your forces, cutting off your access to that system has hardly good policy.

Hope this helps to answer your question. I'll try and look through my books later when I'm home to put some sources in here. This is off the top of my head, so I'm more than open to corrections from those with more specific knowledge.

Side question:

Someone correct me if I am wrong, but wasn't one of the reasons that King Philip IV massacred the Templars was because he was so heavily indebt to them? If this wasn't a common theme, when did this practice stop?

A mix of goodwill and "they didn't". In some cases a ruler would keep a personal banker such as Jakob Fugger in the Habsburg Empire to maintain a constant line of credit. The usefulness of this banker depended on his continued ability to raise credit. We have to keep in mind that banking was a very informal institution prior to the foundation of the Bank of England, and started as a system of lending between close-knit groups of merchants and business associates. For the King, lending was about "getting money now", and a banker remained in the inner circle where he could expect the King's good graces as long as his ability to procure money outweighed the King's mounting financial burdens.

An interesting phenomenon going off of this was the "court Jew". For a long time, Jews represented a majority of stake in Medieval banking due to usury prohibitions. Monarchs kept around a Jew who would raise capital. The position was extremely precarious because a change in King or cabinet could mean sovereign default and the ruin of the court Jew. The same went for ordinary bankers. For a banker, however, being the personal banker of a King carried prestige. Fugger was able to bring credibility to deposits within his bank by name-dropping the Habsburg monarchy: in the same vein, famous banker Cosimo di Medici was able to build the Medici bank after becoming the official banker of the Pope - so despite the risks, lending huge sums to the government could create payoffs in the long run.

Defaults carried their costs for the crown, however. On top of losing a brilliant financier close to the King, the monarch who defaulted had trouble raising more credit. Charles II of England had to force noblemen to give him loans because interest rates were so high from previous defaults, and rarely repaid them.

However, desperation and wars often drove monarchs to bankruptcy, and it would ruin bankers and prove how powerless they were. As an example, the entire city of Genoa suffered economic collapse and impoverishment due to Spanish defaults. After losing its trade empire to the Ottomans, the trade city of Genoa struck rich once more by becoming the "banking city" of the Spanish empire. That was, until Philip II defaulted four times. The economy of Genoa was so crushed that what was once the arguably richest city in the world drifted into utter irrelevance - and it still remained a Spanish ally, proving just how powerless creditors were to collect on debts.

This era of terrible banking started to end with the creation of the first "national bank", the Bank of England. Realizing that they could secure much lower interest rates and stimulate lending in the general economy if they created a trustworthy bank that the government was bound by law to repay, the English parliament created the Bank of England to manage the country's debts. The experiment was hugely successful: the English halved their pre-BoE interest rates, and interest rates continued to decrease even when the nation's debt to GDP was over 200%. Imitations slowly appeared all over Europe, like the Bank of France.

Source: Drelichman, The Sustainable Debts of Philip II

Arkin, Aspects of Jewish Economic History

North & Weingast: Evolution of Institutions in 17th Century England

The answer is that they didn't. It was a big problem for medieval bankers. If you look at the history of the Medici family, for example, they lost a lot of money when several prominent leaders defaulted on their loans, and it helped spark their transition from a banking family into the ruling family of Florence, which held political power but was no longer the banking giant it was in the 1400s.

One other thing to mention here is that the medieval world was one of highly personalized relationships. Most banks were family operations at heart, and they often developed close connections with their powerful clients. Take a look again at the Medici family and their relationship to the Papacy.

By the same token, you could ask, how do banks keep countries from defaulting on their loans today? The answer, again, is that they can't. If a power country wants to default, they have a monopoly on legitimate force within their borders, and nobody can stop them. But it will ruin their reputation and nobody will want to lend to them anymore, or else will want high interest rates.