Turns out that the Cleveland Plain Dealer followed up on this, and managed to reach both of the people involved, who said that both the letters were real. This was good enough for the professional urban-legend checkers at snopes.com, so it's good enough for me.
The general counsel, James Bailey, now lives in San Diego. He confirmed he had written the response. Bailey's web page at the consulting firm where he now works says he was an executive VP for the team, but he had been a partner in a law firm and so it's plausible that he was general counsel too, as the letter states. (Although the letterhead is for the "Cleveland Stadium Corporation," not the team, according to the web page Bailey led an effort to get the team a new stadium, and so could have had separate letterhead for that purpose.) Bailey also confirmed he had copied Art Modell, the team's owner, on the letter, which might not have been the best idea. "I should have been more cautious," Bailey said. "After I wrote it, I heard about it right away from Art. He said something like, 'What the hell are you doing?' He was not a guy lacking passion."
The complainer, Dale Cox, has since moved to Idaho but is still practicing law (and, he says, is still a Browns season-ticket holder, which might show his judgment has not improved). He told the Plain Dealer he wasn't mad about the response and that in fact he "thought it was pretty cool." Whether he's remembering that correctly or just doesn't want to seem like a sore loser now, that's the right response. He also claimed to have "used that letter a couple times myself since," but if he did, he did not provide details.
I came across a couple of posts suggesting that Mark Twain originally came up with this idea, but if he did, I couldn't find it; and it would probably be public domain and/or fair use anyway, if like Mr. Cox you wanted to use this yourself.