The power of the monarchy could be extended longer but only at George III levels of influence, nothing more. George and his advisors did their best to have as much influence as possible, and modelled him as a Patriot King above partisan politics, changed the system of patronage so it was in the hands of whoever was PM rather than those who knew the system best (i.e. the Pelhams), and would thwart potential PMs he disliked.
However, ruling directly like a US President is simply off the cards. Through his best efforts, he was the most powerful political stakeholder in the system but still a stakeholder. The UK is clearly a parliamentary system at this point and George had to operate through that, using patronage and threats to cultivate his own faction and make alliances with others. It is also a parliament with the vast majority of MPs being Whigs, formally or informally. This was defined as respecting the constitutional settlement of 1689. The end of the Jacobite movement in 1745 actually makes the monarchy weaker, because it ends the sense of "better the Hannovers than the Stuarts".
As mentioned, what could happen is a longer continuation of George III's level of influence. IOTL the monarchy continued to decline in power to be virtually fully ceremonial by the beginning of Victoria's reign. If you avoid George III's madness that helps, and you would also need more activist sons who are both politically focused and leaning more autocratic than OTL. However, you are fighting a general dynamic towards being a more and more parliamentary and democratic country. Kings can maintain or diminish their power, but it's very very hard for them to increase it. That means a trend towards ceremonial monarchy sooner or later.