The scenario I'm imagining would involve the Qing (or Yuan Shikai's family, or someone else claiming the throne) staying in power but delegating all the actual power to elected officials, like the UK.
The short answer to this question is a big fat YES. Constitutional monarchists were in fact the dominant reformist faction in the Late Qing, and remained hugely influential into the early Republic. Before 1900, the dominant radical figures in the debate over the long-term state of the Qing monarchy were undoubtedly Kang Youwei and Liang Qichao, two Western-influenced (but, notably, not Japanese-educated) intellectuals, and the latter remained hugely influential in the days of the Beiyang Government (1912-27) as the leader of the more 'moderate' caucus of the National Assembly.
The idea of curtailing the authority of the imperial state was not a uniquely Late Qing phenomenon: advocates of fengjian ('feudalism') in the 1660s had argued for devolving authority for local administration to local elites, and the clamouring for local authority by extra-bureaucratic gentry elites had been a long ongoing process. Philip Kuhn's classic argument is that they had become increasingly successful in this endeavour following the military disaster of the White Lotus Rebellion in the 1790s, which encouraged the establishment of decentralised military forces that only became further expanded in response to the crisis created by the Taiping War in the 1850s. Feng Guifen, poster child of the Self-Strengthening Movement in the 1860s, had argued for the creation of locally-elected sub-magistrates who could serve as a grassroots check on corruption.
But constitutionalism proper as both a concept and a political movement emerged very much in the 1890s, as groups increasingly dissatisfied with the Qing state – especially so after the defeat to Japan in 1894-5 – pushed for a political consensus that favoured the Han Chinese provincial elite, rather than the increasingly Manchu-dominant imperial centre. In many ways, the constitutionalists were simply asking for (to impose for a moment the idea of the separation of powers) greater elite involvement in legislative procedures that mirrored the provinces' increasing executive autonomy. While there was not ever necessarily a clear, broadly agreed-upon statement of purpose, many of the crucial proposals that came through included the promotion of a greater number of radically-minded Han Chinese to positions in the central bureaucracy, and the creation of provincial and national assemblies based on limited (property-based) franchise.
The high water-mark for the constitutionalists was undoubtedly 1898, not only the Hundred Days' Reform in Beijing during the summer but also the provincial reform movement in Hunan, which had been ongoing since 1896 but reached a climax in June. I wrote about this quite extensively here and won't go into too much depth here again, but the salient points of the pre-1899 constitutionalist movement are thus:
Firstly, constitutionalism was not purely about political reform. The constitutionalist intellectuals presented themselves firstly as economic, social and military reformers, who seem to have used this as a sort of 'gateway drug' to their more, shall we say, pernicious political agendas.
Secondly, constitutionalism revolved around the promotion of provincial power and a reduction in central oversight. This was not necessarily out of a desire to weaken government – far from it. Rather, reducing the amount of interference from central bodies in the management of province-level affairs was aimed at strengthening the overall efficiency of government and ensuring that China remained geopolitically competitive.
Thirdly, for some, constitutionalism had a distinct ethnic character. Whether out of apathy or antipathy, provincial constitutionalists like Kang Youwei, Liang Qichao and Tan Sitong were proposing the end of the old system of Manchu-Han 'dyarchy' in favour of a Han-dominant consensus. This alienated many Manchu leaders, who were quite understandably concerned about what would happen if the declining influence of Manchus in the provinces were to be replicated in the imperial centre.
Little wonder, then, that when push came to shove in September 1898, conservatives and moderates, Manchu and Chinese, banded together behind the Dowager Empress to metaphorically arrest the growth of radicalism and literally arrest the reformist leaders.
Still, the constitutionalist faction hung around. Kang and Liang avoided arrest and toured round Asia, particularly India and Japan, for some years, continuing to write inflammatory material in defence of the Guangxu Emperor, now under house arrest, and continuing to suggest radical political and ethnic reform. Liang in particular seems to have developed a fascination for Social Darwinism, and pushed heavily for instituting policies of not just Han-Manchu desegregation, but indeed of compelled miscegenation. While the radical constitutionalists played little role in the New Policies that followed the Boxer Uprising in 1900, there is much to suggest that constitutionalism – in some cases in a more moderate, less anti-Manchu form – remained an attractive prospect even at the time of the 1911 Revolution, especially given the death of Cixi in November 1908.
The Zaifeng regency of 1908-11 was a period that was not without promise, as the final pieces of the nine-year New Policy programme fell into place. The continuation of the empire's military reorganisation was accompanied with attempts at industrialisation, albeit heavily foreign-funded, and crucially there was a commitment to empire-wide political reform. In 1909 the first provincial assemblies met; in 1910 the first National Assembly. This was not, however, to be the end. The Assemblies were to be stepping-stones to the formation of a national parliament as the legislature and a modern-style cabinet to lead the executive. By 1911 the Qing state was very much heading in the constitutional direction. There were, of course, controversies. Among the most salient of these included Zaifeng's declaration in November 1910 that no parliament would open until 1913, and perhaps most damningly of all the announcement of the cabinet's membership in May 1911, which contained nine Manchus, including five imperial clansmen, and only four Han Chinese. Worse still, the Prime Minister, Yikuang, was decidedly unpopular, having been the subject of a failed assassination attempt in the autumn of 1909. While the form of constitutional monarchy was being assumed, many doubted that the actual substance was going to be there.
The whole constitutionalist project was more or less completely derailed by the outbreak of revolution in October 1911. The outbreak of revolt all across China – most prominently in Hubei, Hunan, Shanghai, Sichuan, Shanxi and Guangdong – presented a fundamental, existential challenge to the survival of the Qing state. Yuan Shikai, who returned from his semi-exile to command the Beiyang Army late in the month in response to the outbreak of the revolution, was made Prime Minister in November with authority to negotiate a settlement with the revolutionary elements in the south. But Yuan was not simply negotiating with committed republicans like Sun Yat-Sen. In fact, the discussions were dominated by comparatively moderate interests such as those of Li Yuanhong, the somewhat hastily elected leader of the Hubei revolutionaries. Li, like Yuan, had previously aligned with the more moderate constitutionalist faction, but the experience of the revolution had evidently pushed Li in a republican direction. Yuan, however, still believed it possible to negotiate for a constitutional monarchist settlement rather than submit to the revolutionaries' republican demands.
Two problems arose, however: firstly, his negotiation strategy was quite poor. He didn't pick a particularly good negotiator, and his intransigent opposition to the revolutionary demands as well as continued insistence on the viability of a reformed monarchy were not particularly endearing either. Secondly, his window of opportunity was rapidly closing. While he had achieved sufficient military success to drive the revolutionaries from Hankou and Hanyang, the Hubei capital of Wuchang remained in rebel hands, and the various rebel factions, not just in Wuchang but also in Shanghai, Nanjing and Guangdong were beginning to coalesce during the ceasefire. If negotiations broke down, civil war would likely resume and with far less certainty of northern loyalist victory.