Have you figured out what your exact costs are for everything? What are you including in "uncovered expenses" for top surgery? Time is on your side. The courts in most places are very slow. This is especially true if you have to physically appear in court to have the name change processed. I didn't, and it took about a month. This is what I ended up doing monthly (started medically/legally transitioning in December, top surgery was end of April):
December - Met with lawyers to draft paperwork, got letters from doctor for document updates
January - Paid for name change and ordered copies in advance
February - Paid top surgery consult fee and deposit
March - Paid for new driver's license with updated name/gender marker, booked hotel room for top surgery, got prescriptions filled
April - Paid balance for top surgery
May - Paid for updated passport
It didn't cost me anything to update SSA, college records, credit cards, insurance policies, etc. It maybe cost me a total of $5 in postage to mail things to people that needed them, and maybe another $20 in parking fees (I live in a city). My total investment for updating all my records was under $250.
I actually didn't spend money on other little expenses for top surgery. I made an Amazon wishlist and shared it on Facebook. I didn't feel comfortable asking for money for the surgery itself, but a lot of my friends/family wanted to make sure I had the things I needed. I got all the necessary goodies like bandages, Neosporin, gauze pads, medical tape, straws, stool softeners, baby wipes, etc. I also added Amazon gift cards on there, and several people gifted me those as well. I highly recommend doing this.