There's no legal reason why you can't start a degree at 17 as far as I'm aware.
Disclaimer: this is my perception based on people we know, and not scientific. Years ago very talented performers could train at 16 and ended up on the pro theatre circuit. Now I think that sort of success is more common with degree grads from top schools who are more like 21/22, and it's harder for those who do a diploma 16-19, especially at less prestigious schools, to get much beyond holiday parks, being cruise dancers etc. That may be a sweeping generalisation, and of course there will be exceptions, and it very much depends on aspirations too, but worth thinking about. I hasten to add that holiday parks and cruises are VERY valid career paths - not looking down on them at all - but it just depends where your DD wants to end up.
Whichever you choose, it is long hours. A big commute on top might be too much. The Laine foundation could lead to degree at Laine or other very good schools. A diploma from Performers will no doubt be good training, but then you have used up your student finance and you can't then progress to a school that might open up more doors.Although Laine is known for its dance, under new management they are, I think, trying to provide proper triple threat training. Performers takes people who are mainly dancers - have known people go there who are really not singers - and realistically it will train them for dance work. Again, it depends whether your DD is mainly a dancer or wants to be a full triple threat.
I haven't intended to offend anyone - both schools very good at what they do, but they are offering something quite different. At 16 you have time on your side and I'm not sure I would rush into a 3 year commitment unless I was absolutely sure it was the right fit.