Is a cosmetic logotype on an outer copper layer a bad idea?
Nope, there really aren't much downsides, especially with solder mask openings.

The solder mask is electrically quite insignificant. If you'd have the opening on an impedance controlled trace it would have slight effect, but you wouldn't have it there anyway, because it would be ugly. If you have ENIG finish on the board, there's added cost, because there will be gold on the logo as well. You can avoid the cost by placing the logo on a place where there's no copper under the solder mask, but then it isn't as nice and flashy.

You can't make fancy or small graphics with solder mask, because thin lines don't stick to the board well.

Etching the logo to copper might have some effect on EMC, depending what's underneath. If there's a ground plane on the next layer, then you most likely don't have any issues. The modern etching processes are quite impressive so with copper you can get a bit more details than with solder mask, but it might come with added cost, if the board doesn't otherwise have thin traces on that layer. And there's also the added cost of added gold with ENIG finish.