Those hyper-intelligent pan-dimensional mice sure are good at getting humans to do all the work to cure mice of all disease and aging.
Sure, if you *cap* the population at 150 during transit, and don't allow multiple pairings within the same generation of course you're going to kill the genetic diversity.
However, if instead of a generational ship we were talking about hibernation until arrival, 150 is enough to begin a genetically viable colony. How do you avoid the risk of inbreeding? Simple: no cap on the number of children, but no full siblings allowed. Encourage as many different genetic pairings as possible.
I will also note that the rights holder recently released an on line version of their board game thats playable on line. However it is very specific and limited in options. It seems more then coincidental that we've received this notice now that their own game lobby is on line. Can our application take refuge in another country? Is there a way to keep our application alive since it is significantly more feature full?