Build Advice: Natural Attacker

Hey party people.

Making a backup character, and I saw the 'Mutation Warrior' archetype for the fighter and thought "well that's neat" and then immediately decided multiclassing was just BETTER.

So here's my plan:

2 levels of alchemist for: A few 1st level extracts (shield, enlarge, long arms) Discoveries (lots of good ones don't have level requirements, notably for us tumor familiar, vestigial arms, feral mutagen, spontaneous healing). Get one at second level, but then the 'extra discoveries' feat let's us grab the rest Mutagen

X Fighter. Specialize in natural weapon group and buffing AC.

Tiefling can get me claws, and combining with vestigial arm discoveries and feral mutagen, I now have 4 claws and a bite attack at a fairly low level. One vestigial arm can just hold a shield until I get the other claws going, and with Long arms, all 4 claws have a 10ft reach.

My only real concern is that this is basically the end for the build. Once you've got your 4 claws, your only other advancement comes from fighter progression (which is nothing to sneeze at).

Anyone have thoughts? I just think natural attacks are fun, and thought this might be viable. I'm sure a beastmorph/vivisectionist alchemist would come out on top eventually and have a lot more versatility, but I wanted to try something different. I'm open to other builds as well.