Monoid theorist
Category theorists are mostly conceptual mathematicians that use category theory as their main tool of choice; it is uncommon to see category theorists that are interested in categories the same way that a group theorist is interested in a group.
If this is what we want to transmit, the name “category theorist” may not be the best: we work with categories, true, but we also work with bicategories, premonoidal categories or profunctors. If we want to say that these are all concepts derived from a more primitive one, that one is possibly “monoid”, more than “category”.
Monoids are the simpler algebra of composition, category theorists study composition; we should probably be calling them “monoid theorists”.