So, I have a tamed pet and I already used a lesser mutator, didn't get any visible traits. There's no other way now to give it more traits, is there? Wondering the same about my starter, if there's any way I can add to it's design.

Unfortunately, once a single Mutator of any sort has been used on a pet, you cannot use another on them.

Lesser mutators are very much a roll of the rng, you just got unlucky with it giving you a carry (at least with a vis you don't like you can potentially change it later using an RTD or a trait scroll.)

At this time starters cannot be changed (although it is a possibility that some time in a distant future there will be a way to alter/regenerate them or something to that effect, not anytime soon)

On a Mutated pet you can use Dyes with no limit to change a pet's colours, or RTDs/Scrolls (keeping the same TP count by swapping, rather than gaining more TP), but that's it