If someone decided to make a trademark today out of what had been your personal domain for years and they showed up, offered to promise not to sue you for a fee of $1 how would you feel about giving it to them?
Don't get me wrong, $1 is nowhere near worth going to court over. But come on... do you want to pay me a dollar not to market something under the name j-beda?
You are right that I would probably feel insulted, but if I responded to their letter saying "we are not in the same business, there is no infringement, I don't need a license" that in itself would probably satisfy their need to defend their trademark. If their response was, "sorry, it looks like there is infringement, you really need a license" I might actually consult a lawyer.
The offer could be more of a cross-liscencing deal with no money needing to change hands.
Remember, this is under the assumption that there really is some actual need to defend the trademark. If some company actually spent money on a stamp and a polite request to make sure the "Juvenile Base for Egregious Dumb Asses" was not confused with "j-beda", I might be open to a disclaimer on my website.A rude demand with threats from random bed manufacturer who wants me to just turn it over without question, not so much.