Is there an actual regulation on the wear of the black fleece cap in PT indoors?

Per DA Pam 670-1, the fleece cap is an authorized accessory (and AR 670-1, para. 10-2, “note” directs to the DA Pam for accessories). The actual wear of the fleece cap is a WABC, so check the local guidance.

Here is your answer, IAW at DA PAM 670-1 PARA 18-12 c "The fleece cap is worn with the physical fitness uniform or the combat uniform, as prescribed by the commander." Your installation command should have a policy letter out regarding the wear of these items. We in Alaska have even further guidance having the Balaclava as well. If you do not have specific guidance covering this at your unit, you have to look at this in a manner fitting this accessory. You are correct in stating that it is not listed as headgear, rather listed as an Accessory. But the Beret is also listed as an Accessory and then the regulation goes on to state "..the standard headgear worn with the Army service uniform" So we have to apply some common sense and correlate that maybe, just maybe, the Fleece cap is headgear though not explicitly stated. But when in doubt grab your 1SG and go ask your Company Commander how he prescribes the usage of that "Accessory" and tell him CPT (P) Daily sent ya! But I would recommend you keep it low and protect the WOLF pack.

Yes sir there is. AR 670-1 and DA PAM 670-1.
On your second point, how did you come to that conclusion sir? The reason I ask is because the referenced publication does not have a section dedicated to identifying what is and what is not considered headgear, the DA PAM does.