Even if they want to give it up, the agreement will likely have representations and warranties (see part about any known/future claims) which OP should not just sign without an attorney reading it over. The agreement will surely be written in a way that is very favorable to the other party.

It isn't that simple. The burden of proof and the cost of filing suit is entirely on the people who own the .net domain. You can't lose a domain based off of a threatening email.

I have received threaten letters before and have responded with "I dispute these claims. Please do not contact me again in any manner" and I never heard from them again.

I don't know enough about the company sending the threatening letter here, so I can't say if that would work or not here, but it is very possible you don't need a lawyer.