There’s no reason to be afraid of Fiverr closing your account unless you do anything that’s against the rules, declining unreasonable cancellation requests certainly isn’t against the rules.
Don’t worry, keep calm and polite, deny the cancellation requests and explain to your buyer that you delivered everything you had agreed on/your gig specified you’d deliver.
In case you feel like continuing to work with them (I guess probably not) and can even do that job he wants done in addition to what he paid for and you delivered, you can offer to send them a custom offer for the additional work.
You can reject his cancellation request until he gives up or goes to support. If he goes to support, they might tell him the same as they told you, or they might cancel, in which case you’ll have to prove to support that you did deliver what he bought.
But only start worrying about that if it actually will happen, maybe it won’t, the buyer might eventually understand that they’ll have to make do with what they paid for.