IMO - In My Opinion
There are two types of FSA's - dependent care and health. In a dependent care FSA, IRS regulations do not allow you to claim money until it has already been deducted from your check. (Query - had the full $3000 already been deducted? It should have been divided into however many paychecks there were left in the calendar year from when you made your election, and one portion deducted from each paycheck.)
On a health FSA, while the deductions are made the same as I've described above, you can claim the entire amount you elected for the year, the very first day of the plan year if you want to. You then essentially repay your account as you go through the year. However, if you've claimed more than has been deducted from your paycheck when/if you leave, that's just too bad for the employer; they can't ask for the money back. Health FSA's are also use it or lose it - if you don't submit your claims on time, the money is gone. The law was written with the use it or lose it provision being both ways on purpose.
By the way, I'm not saying that you should have known what the policy was. I'm saying it was your responsibility to ask rather than assume. You decided that you knew what the rules were and you were wrong. You were wrong about March 31 being an IRS deadline and you were wrong that all plans have the same rules on termination. But if you had asked you would have been told (and if you had asked and had been told wrong, you'd have some legal justification on appeal). You didn't ask; you got it wrong; you didn't read what you were sent, and you ended up on the wrong side of IRS regulations. The employer is in a much stronger position legally in denying you, then you are in saying, "No one monitored my account and sent me a follow up so I wouldn't have to pay any attention to dates and deadlines". Because one thing I will promise you, if you try to claim you weren't told, they will be able to produce evidence that you were. Because, and this is what I was trying to get at, you cannot possibly be the first employee to leave the company with an FSA balance, and somehow all those other employees got their claims paid. So either the information was written down in a place they had access to (which means you also had access to it) or they had the sense to call and ask. It would surely incite remark (and IRS investigation) if no one was getting any claims paid after termination, so I think it's a fair bet that you're the exception rather than the rule.
I know you are looking for someone to say, "Tell them this and they'll have to give you your money back". I've already told you the best possible way you can do that and you don't want to do it that way. Please pay attention here. I'm going to say this once more.
You are in a bad position here. The law is on your employer's side, not on yours. For your employer to "help" you would involve their having to violate the law on your behalf. They have no incentive to do anything to help you in this instance. They are legally in the right and you are legally in the wrong. There IS no magic wording that you can use, or law you can invoke, or arguments you can muster, that will guarantee you a win. If you are determined to appeal, all you can do is make the best plea you can, NOT by putting responsibility on them, which is only going to put their backs up, but by taking responsibility on yourself, which is where it belongs and may make them look at you in a kinder light.
Now, let's talk about your health insurance, since you say you did call about that. When did you call, what were you told, what was your last day of work and what was the last day your insurance was in effect? Did you receive your COBRA notice and if so, when?