You do not need weights or a BCD to dive in warm water.I was told that I don't need a BCD, tank and half the weights. Also, you drag the kayak with you on the tow belt, which also serves to hold the dive flag. My primary interest in scuba right now is to hunt lobster. I felt like all the gear I had was a major hindrance on maneuverability and comfort underwater as opposed to skin diving.
I do a lot of cold water diving in swim trunks carrying an AL80 cylinder (in my hands; your preferred attachment type may vary) and no BCD and no weights. Two pounds of freediving weights would be comfy, though.
Hence, you could carry a tank and a regulator and not much more. You'd need to attach it somehow, though. You'll want to have both hands free for lobster hunting.