Legal Question in Real Estate Law in Texas

There are a few issues going on here.

Even though you got rid of the pet, the complex has a legitimate claim that you owe a pet deposit (and maybe some other fees listed in the lease). You had the pet in the apartment at some point in the lease term and that is enough.

I am not confident in the landlord's ability to re-categorize your rent as a deposit. If you paid rent then you paid rent, but the deposit is still outstanding. So either way you still owe the complex money. From that standpoint, it may not matter whether the money you paid went first to the deposit and second to rent.

You can be evicted because you violated the lease by bringing a pet into the apartment without reporting it (assuming your lease says you must report pets and pay the deposit plus any other pet-related fees). They can evict you for not paying the deposit although a judge is going to be more forgiving towards you for not paying a deposit right away (unless it is obvious you could afford to pay it) than not paying rent on time. So that is probably why the landlord is trying to categorize your payment as deposit rather than rent. However, it should be obvious from the date it was paid that the check was rent.

You should stick to the facts that you paid rent and not the deposit. Ask the manager or whoever you have been dealing with if you can break up payment of the deposit over the next couple months or whatever they will negotiate. It would not be a bad idea to put your request in writing. They may not be willing to negotiate because they want you out. So you may have to try to come up with the money to pay the deposit this week or prepare to appear in court to defend against the eviction.