I would make her plead her case for why transferring to that school (without him being the catalyst) would be good FOR HER and her career aspirations and the major she has. She needs to put down on paper the puts and takes of credits, transfer fees, how things will get paid, cost of tuition, etc. Then if she really wants to go there, prestigious label aside, I would allow her to go. All things being equal and she will probably do well, etc. But if I were paying, I would insist that this be the last transfer. She goes or she doesn't but she has to want to go there FOR HERSELF. If she were going on her own dime, I'd be less invested.
Also, I would encourage her to do things like get involved in her school, get a PT job, volunteer, or talk to the counselor. It is one thing to be homesick or unhappy but if she's having breakdowns daily, that may be something she needs additional help for. Depression is something to consider. Sophomore year was a tough year for myself and my SD and we really had to work to get through it and keep it into perspective. I saw my college counselor, because that was the year I had a death in the family and I was struggling. I ended up taking fewer credits to get through the first semester. I would encourage her to take action vs pining and evaluate at the end of the semester. If this is their first year of college, it would be especially important for her to take some time to figure out if it's just a change of place she needs to get used to (so many frosh are homesick) or real change of heart about her choice.
As a side note: my stepson broke up with his HS GF about a month into college apart BUT six months apart from his current GF doesn't seem to have negatively affected their relationship. Part of it is maturity and part of it is the relationship itself. Long distance is hard, but if it's truly worth working on, the people will make it work. My stepdaughter is currently dating a guy who is local to home but not to school, since he graduated. They don't see each other often now that he is working and she's on campus, but either it will work or it won't and I would NOT encourage her to quit or transfer to be near him when she has just a year and a half left. She made her college choice long before she broke up with her HS boyfriend, and she made that selection for herself, vs for anyone else. She has the long-term goal of owning her own business some day and now and then we've reminded her to keep her eye on that when she is frustrated.
ETA: I think you have a good plan. I also agree that if she has not seen the campus, then it would be bad to transfer. My SD visited many schools and while Hoffstra gave her a good package to go, it was not her vibe. At.all. So many things need to be considered. Also, if she has to fly to get there, what would it mean about visits home? If she is struggling emotionally as is, would the distance be a factor if you needed to visit her or bring her home for a holiday? I know people who did not go home over short breaks. They stayed with friends off campus because it cost too much to fly. I hope she rebounds.