I’m not sure I entirely understand the question. You want a motion menu where the buttons are animated, but after they run out of video you want them to be still, correct? But then you said that you don’t want to jump to a still menu. Does that mean you have other animation in your menu besides the buttons that you want to continue playing?
The simplest way I can think of doing something like this is to build the animated buttons into the menu background (rather than letting Encore create them) with After Effects or Premiere. After the last frame of the animated buttons, replace them with the still buttons that you want and extend their duration for however long you want. To make the menu last indefinitely, though, you’ll have to loop back to the start of the still buttons at some point. If you’re using audio, you’ll probably want to adjust the loop point to match the audio. If necessary, make a version of the audio with a second instance (so that it repeats itself), and then only loop back to the beginning of the second instance.
Otherwise, what you’re asking is not possible. The menu has to loop at some point, or you have to make a jump between two menus- the animated one and a still version.