By “rat ulcer” do you mean Rodent Ulcer? If so, you need to be aware that these are from an autoimmune disease that is only controlled with steroids such as Depo-Medrol, and that they can occur inside the cat as well as on his lip. When Rodent Ulcers occur inside a cat, the cat may not be able to eat or swallow, or may suffer something like Irritable Bowel Syndrome. They are also called Eosinophilic Ulcers.
They are quite serious and cause quite a bit of pain. Your cat needs to see a vet.
As to inexpensive vets, ask the local Humane Society or ASPCA or any no-kill shelters in your area for recommendations. I have been told that the VCA Lindley Animal Hospital is relatively inexpensive in your area.
Supposedly these two are also less expensive:
Animal Clinic of Farmer’s Branch
14021 Denton Dr
Farmers Branch, TX 75234
Phone: (972) 247–2221
Pet Vet Animal Hospital
12801 Midway Rd # 210
Dallas, TX 75244
(469) 522–5100
From personal experience I have found that vets in the poorer parts of town often charge less, too, so you might think that way.
You also might see if your parents would consider CareCredit, which is in essence a vet services credit card but with convenient monthly payment options, no up-front costs, no prepayment penalties and no annual fees.