Noooooo, this is a bad idea. This is how students get deep in debt without having an exit strategy.
If your mom can pay $100K, that’s about $25K per year. You can borrow $5.5K your first year, $6.5K the second and $7.5K in years three and four. So without any aid, you need to go somewhere the total yearly cost of attendance is around $30K.
UB and Manhattan are clearly within that range. You seen to prefer Manhattan, but UB is also a great school and a good engineering school. I agree it’s bitter cold, but…it’s only 4 winters! (I agree with @crazed - I wasn’t that far north but I was in the Northeast for the last 6 years, and the last two winters have been bitterly cold and harsh. But the ones before that were kind of mild/not so bad.) Virginia Tech is the next cheapest option, but you would need to find a way to come up with an additional $6K a year. Virginia Tech would be the best combinaton of value and academic rigor/reputation and if you could come up with the additional money, that would be the best choice.
GT and UT-Austin are too much money.