I don't like the idea to go to court, but if you do, here are some advices:
1. You really need to get a copy of the forms signed to make sure what you agreed to. If you made a mistake as to the security deposit, you will lose since a unilateral mistake does not render the contract voidable.
2. This case is obviously not within the statute of frauds so that an oral agreement is as valid as a written one, provided the forms you signed were not the contract, as you said. Parol evidence rule does not apply since it is an oral agreement.
3. Other extrinsic evidences such as your credit history are not going to be admitted. They are irrelevent. The only thing you can do is to prove that there is no meeting in mind, thus renders the contract unenforceable. To do that, a copy of the daycare center policy would be helpful to ascertain whether it is its common practice to collect security deposit. Other parents could be your witness if they were not required for security deposit. If the daycare center can show a proof that it is their policy to collect security deposit, then you will lose.
4. There is no saying that you can not go on appeal if you lose. You can go to the US Supreme Court if they determine your case is important to the development of law. Normally nobody would appeal for such trivial case. In a small claim court without jury, judge will decide based on what he thinks it is approriate. Unless he made an obvious mistake and the mistake was substantial so that it affected the outcome of the case, appeal would be denied.
5. Evidence is key. You are the claimant and you have the burden of proof. If you can not prove by predonerance of evidence, the defendant does not have to even say a word to win the case.
6. The outcome, normally, you get your money back, less the 5 days, the agreement is void and you have to negotiate with the daycare center again for future arrangement, IF, without any convicing evidence. The purpose of equity law to prevent unjust enrichment or impoverishment.