Does a direct supervisor have the authority to disapprove my transition leave request?

EXCELLENT point and Spot ON!
It normally takes several days to get that Clearance Form done. CIF alone is a freaking nightmare. Also the if there are any "hiccups" in the out-going Physical, it could trigger other actions/events that must happen prior to ETS.

ACAP is where people learn how to protect themselves while they are still in, as they transition out. Absolutely should not happen while TDY.

This is usually the last "game" you'll have to play with your unit. Transitioning soldiers should be placed into a transition unit, platoon, squad etc. This usually doesn't happen because most units typically could care less about transitioning soldiers (which is sad, but true). If I were the first line supervisor, I would simply refer transition type matters to the S-1 (take the soldier there, etc). I would NEVER make a decision or try to delay any matter like this. I would simply be a "facilitator" or support. But again, folks in the Army these days simply don't have a clue about how to lead "you could ACAP while TDY or just sell your leave and work up to your last day". There should by systems, policies, procedures in place for things like this but- usually there is not. It's a tangled web of a mess created by regulations that leaders don't understand "commanders discretion" or "mission requirements" etc

What was the tasking for?

so glad things worked out for you!!! That's what we here at RP is here for....support and guidance. shame on your 1st line....I assure you, not everyone in a position such as her (supervisor), act in the same manner.