I totally disagree. Look at the letter OP had proposed writing:

After our discussion I reviewed my contract, only to find that Camp Name is in breach. Both parties must agree in writing to each change in deadline and Camp Name has repeatedly extended my deadlines and my time on this project with no agreement in writing and without updating the contract. I would like to remind you that Camp Name does not have copyright to anything I have delivered to that point due to non-payment of the terms in full and contract violation. ...

That is a terrible way to communicate to a client. Not even lawyers write like that to each other. If OP needed changes in writing, when changes were requested by phone, OP should have asked to have them sent in writing. OP has done nothing to uphold that provision of the contract either. And then what OP says above about copyright is almost certainly a misunderstanding of contract law.

If I received the above letter from OP I would be offended and I would think OP to be a total amateur. And if I gave that letter to my lawyer he'd laugh his ass off.

And most importantly, nobody gives a shit. It's not like on a $600 contract someone is going to listen to some vendor bitch about some provision somewhere. I'd just fire them and find someone else. They can take me to court for the money, we both know they won't, and I'm not even sure they'd win since OP didn't deliver the agreed upon product. OP would probably get a judgement for $300 at best after spending $300 in court filing fees.

My point was OP should talk to them as a human, like humans, about what they agreed to. Trying to speak lawyer and accuse them of breach of contract is an absolutely terrible way to manage a business relationship.