Then this happened. And if there's a way out without penalty, we'd like to find it.
The remedy for a lack of clarity in a contract is not for one side to unilaterally void the entire contract. Instead, the two sides fight it out over the correct interpretation.
Given that (1) you can afford to pay, (2) the other side isn't backing down, and (3) the lease has an integration clause, in your shoes I would grit my teeth and pay. And consider this a learning experience. The time to be careful and assertive is when you are in the driver's seat, when you still haven't done what they want you to do. Before you sign the lease, and even better (if the application fee is large) before you submit your application. Insist on seeing everything in writing early enough to reject comfortably without worrying about temporary homelessness.
The problem with fighting something like this now is you can't really sue them. Instead you have to withhold the money (with clear written notice that you believe you don't owe it) and wait to see what they do. This exposes you to the risk of an unfairly damaged reputation. They can and probably will report your rent "shortfall" to the credit bureaus. If and when they sue you, that lawsuit enters the public record. Even if you win or the suit later gets dropped, some future prospective landlords who do background checks may be scared off. And these people may well give you a bad reference if/when called by future prospective landlords.
You could also consider paying the couple of hundred dollars a consultation with a landlord-tenant lawyer probably costs, to get a more knowledgeable opinion.