Given the “gentle bankruptcy” aspect of a King’s visit, were there any particularly . . . unusual ways that they tried to get out of such visits?

Well, A. it wasn’t easy to get out of them, and B. it actually was a social cachet to host the king, and often honors could flow from said visits, so if you could afford it, it wasn’t a bad way to try to win royal favor. 

However, there are more than few cases of rich people pleading poverty (”my estate without my undoing cannot bear it…my land sold and debts not small; how this will agree with the entertaining of such a Prince your wisdome can best judge.”) and other such excuses, especially towards the end of a monarch’s reign when political incentives start to shift.