There are quite a few sites that'll make you custom mugs. I've used Zazzle in the past and they were ok. Not especially good, not especially bad, just ok.
I am not a lawyer. But, if you go down the route of following their instructions, I'd suggest you should follow their instructions completely, including deleting the account. Otherwise, they could claim further down the line that you failed to follow their instructions.
Also, you should probably tell them you've done it, preferably with a screenshot and timestamp, just so they know you're keeping records. ;)
Also, even if that were an issue, please forgive me but I'm not entirely convinced that the problem of people posting comments that are lacking in nuance could be solved with a tool that has no nuance.
There are quite a few sites that'll make you custom mugs. I've used Zazzle in the past and they were ok. Not especially good, not especially bad, just ok.