I'm not sure how, other than Inman completely giving up on greeting cards, or somehow completely separating it from The Oatmeal.

Third option: Inman pays Oatmeal Studios -- either a consideration for using their trademark, or buys the trademark outright.

Fourth option: Inman licenses his content to Oatmeal Studios, who then handle the greeting card side of the business and pay him a royalty.

(I have no idea of the relative turnover of these two businesses so I don't know if a licensing option is feasible.)

Fifth option: Inman keeps making greeting cards, but brands them differently.

Sixth option: Inman mobilizes his zombie hordes and obliterates all evidence that oatmeal studios ever existed...

He could just name his greeting cards something else. From what I'm reading the other company doesn't have a problem with the site, just the cards (and it does appear to be a reasonable case -- they've been selling "Oatmeal" greeting cards for 35 years; quite possibly longer than Inman has been alive).

I just took a look at the Oatmeal Studios website and some examples of their greeting cards. They don't seem to be in the extreme 'Grandma/kittens&&puppies' sector of the market, so I wouldn't automatically assume they would consider Inman's sense of humour detrimental to their brand. This might be settled through financial consideration.

Then again, without direct knowledge of the principals, it is impossible to predict how this might play out.

I agree with the first two, not so sure it will resolve amicably though - perhaps.