Constructivism is significantly more fundamental than AC. AC is notable for being famously perplexing, less foundationally perplexing. For a more graspable investigation of the perplexities and arbitrariness of our choice of foundational set theory, take a look at Vicious Cycles where the author investigates what happens when set theory does not assume a terminating world (and thus introduces a lot of things that we use in CS often, try to model on sets, and are left dissatisfied because we really wanted costa, not data).

From there please take a deep look into topos theory and you'll discover again that AC is just one choice of properties that make for a meaningful "foundation of mathematics (of a kind)" and a not particularly remarkable one at that.