In my opinion, the best case scenario is that strong talk of UConn going to the XII spurs the B1G (option 1, especially as the ACC may not survive over the long term) or the ACC (option 2) to grab UConn now.
But, if UConn does join the XII, where do UConn's Olympic sports go? The non-revenue sports, such as baseball, softball, track, women's soccer, etc. would have a devil of a travel schedule. Field Hockey I assume can stay in the Big E and ditto for both hockey teams in Hockey East. The XII does not sponsor men's soccer (West Virginia is the only XII to play soccer and they play in the MAC), so that team would have to find a new home - MAC, stay in the America, Big East?
Many sports would not find trouble at all.
UConn sponsors the following sports that the Big XII does not: men's and women's ice hockey (who would stay in Hockey East), women's field hockey and women's lacrosse (who would stay in the Big East), men's soccer (who would presumably join the Big East). The Big XII sponsors three sports that UConn does not at a varsity level: equestrian, women's gymnastics, and men's wrestling (UConn could be easily convinced to, if not elevate its equestrian program to varsity due to its status as a women's emerging sport and UConn's general goodness at it at a club level, compete against their potential Big XII opponents in the sport: Baylor, K-State, Okie State and TCU).
Football, men's basketball, and women's basketball are also equally irrelevant, as they already would have revenue offsets for the travel due to those sports granting ticket and television revenue.
The remaining sports that both UConn and the Big XII both sponsor:
1) Cross country, women's crew/rowing, swimming and diving, and track and field: the team would compete in the Big XII championships for these sports, but would compete regionally for all of its regularly scheduled competition. Travel impact for these sports: minimal to irrelevant.
2) Golf: the team would compete in the Big XII championships for the sport, and would (because of UConn's home climate) play a predominantly Southern US-based schedule (against a mix of Big XII and non-conference opponents) to begin the season, which is something the golf team already does anyway.
3) Baseball and softball: the teams would compete in a Big XII round-robin or round-robin-alike competition, requiring regular travel to the Southwest and Midwest (and West Virginia); however, because of UConn's home climate, the teams ALREADY play a large portion of its schedule in the South, Southwest and West during February and early March (and a good portion of that is against OTHER Northern colleges who are also playing in Florida or Texas during that portion of the season). No noticeable change to the travel impact.
4) Men's and women's tennis: would play a predominantly regional schedule, and would compete irregularly against Big XII competition. Impact would increase, because UConn would be trading away conference competition against Temple and the "south Atlantic" AAC schools for competition against predominantly schools from Oklahoma and Texas.
5) Women's soccer, and women's volleyball: Impact would increase in a significant but not dramatic fashion (because AAC round-robin competition already sends them to Texas and Oklahoma, as well as Florida and Louisiana).
So between those eleven sports: seven would see no impact, and four would see some.
In other words, the pain UConn would feel is ultimately not a whole lot.