How does inserting 起来 into 请客 as in 请起客来 changes the meaning?
This infix -起-来 is usually considered a variation of the suffix -起来, and analysed as having an inceptive aspect, also called the inchoative aspect. The sense that it produces is "starting to do", "to begin to" etc.

起来 often interacts with 了 to form a particular viewpoint of an "accomplishment" action as having starting in the past, with the focus on the start point rather than as a whole.

It should be contrasted with 着, which does not emphasise the start of the action, but presents the action as one continuous state. Additionally, 起来 is different again from 下去, the latter of which is often called a continuative.