I’m thirding Drew for you, as I think it could be a really good fit for you. Another east coast option, but not as close to a big city, that you may want to consider is Muhlenberg.
A reachier option might be Brandeis (MA).
Have you thought about Oberlin? It’s not in a big city, but it’s not far from Cleveland. Strong liberal arts and an actively welcoming community for the LGBTQ+ community.
I know your preference is for the east coast, but the midwest has some really terrific options that will are likelier to get you closer to your price point. I’m thinking of:
Kalamazoo (MI)
College of Wooster (OH)
Lawrence (WI)
Beloit (WI)
Another school I’d take a good look at is Marquette (WI). It’s more of a mid-sized school than a small one (though definitely smaller than BU and Fordham). It’s in Milwaukee, so definitely a city. From a cursory glance at this Jesuit institution (as Fordham is also Jesuit) appears to be actively trying to be a safe and welcoming space for trans students.
You can’t lump all small Christian colleges (even if located in small towns or rural areas) all together.
Christian schools exist in 3 (4) flavors
like Georgetown, Holy Cross, Fordham, St Olaf : founded by a religious group (often immigrants), they keep religion, philosophy and ethics as essential to a person’s education; humans are seen as having a spiritual dimension and the college will support spiritual growth, but all faiths&atheists are welcome. There are clubs for various religions. There may be crosses or crucifixes on some classroom walls and a chapel on campus. Religion is taught as an academic discipline.
like Notre Dame, St Benedict, Pepperdine: the Christian faith is fundamental to the college’s past&present identity and most students embrace it. Cru, FCA, Newman, YoungLife… are active on campus and questions such as “what are you giving up for Lent” would be common. Religion/Theology/philosophy are taught as academic disciplines. There may be a code of conduct (like “parietals”) but no creed to sign. Non Christian students who are believers (in judaism, islam, sikhism, buddhism…) may or may not feel uncomfortable but won’t be made to feel that their religion is “wrong” and are welcome. Being an atheist may be uncomfortable but being a nonbeliever is fine.
like Biola, Wheaton IL, Grove City, Pensacola, Liberty, John Brown: evangelical/conservative Christian faith is seen as fundamental to everything; learning must be done for Christ’s glory. There is a creed students and faculty must sign, with requirements wrt behavior (which generally include no premarital or homosexual sex and no drinking, but may also include no smoking tobacco, no coffee, no dancing, no R-rated movies, no specific types of clothes or jewelry…) Missions are seen as important and students who aren’t Christian may be welcome but may not feel welcome. Religion is taught as truth or dogma, not as an academic subject, though it may be questioned depending on class. There is a chapel or chapels on campus and attendance is mandatory several times a week.
(The 4th type is a college that used to be affiliated to a Christian faith but really isn’t linked to a religion anymore: Davidson, Muhlenberg, and most Ivies are like this.)