Yes, you can and should be discussing your financial package with Belmont, and ask if there is any way any more money can be found for you. Yes, you can mention the large award NYU offered you as well. You should give it a try. Sometimes, you can get more money when you let an admissions officer know you want to go to a school, especially those private schools without a lot of name recognition.
My son was able to get additional merit money from a small private school, and he did let them know he had been accepted to Cornell University. However, cost for him to go to Cornell, to one of the land grant colleges, would have been less than what the LAC was going to cost. Even with the additional bit of money, it cost more to go to the LAC. They did not match the bottom line cost of Cornell. But they did come up with more in grant money.
In your case, the difficulty is going to be that it costs MORE for you to go to NYU than it does to go to Belmont, from what I can gather from your post. THat issue might be raised by the financial aid officer or admissions officer.
What is your EFC? Most always, schools do not give more money than what will have you paying your EFC. If you have a mixed merit and financial aid award, it’s rare that the merit portion will be raised, if you are at that EFC level. Certain financial aid, like PELL, can not be given on top of scholarship money. It is given for need only. So if you get more scholarship funds so that you are reducing your payment below your EFC, the PELL has to go away. So it can be a tricky situation for the financial aid office.
I feel anyone who is thinking about discussing their award, whether it is merit or need or both, should discuss the matter with the appropriate school officials if for no other reason, but for peace of mind. There have been cases, when the discussion results in more funding. However, do manage your expectations. I have not known any cases, other than outright mistakes that were made, where substantially more money was added. Often it’s some loan. It’s at the tail end of the season, so funds are tight.
Good luck in this endeavor and let us know how it goes.