How do I use a Lesser Genetic Mutator? The item appears in my inventory but when I try to apply it to a pet that has an empty slot, it doesn't appear in the drop down box. Is there something I'm missing?

Genetic mutators, lesser and greater, can only be used on tamed pets that have not been mutated before. Tamed pets are those caught in the wild, purchased from the scale shop, purchased from the puffadore shop, or purchased from the Ny'vene shop. You can tell if a pet has already been mutated as it will have this symbol on its profile:


Also, when using a lesser mutator you may get a carried or visible trait. If you get a carried trait you won't be able to see it unless the pet has been genetically tested.

So I can't use a mutator on a pet I've hatched instead of caught?

If it is a pet you bred (aka. has parents), then no. If you found the egg in the wild, or bought the egg from the three shops I listed, then yes (because those eggs are considered tamed). I took a quick look at your pets, though, and you only have 2 tamed pets and both have already been mutated, therefore you have no pets at the moment that would qualify to use a mutator on.