Is there any Army or DOD policy that prohibits biological male soldiers from wearing a bra while in uniform (IPFU, OCP)?

I don’t know why it would; since it’s not worn on the outside of the uniform who would know?

If you receive any friction from your chain of command, another avenue to explore would be medical/behavioral health. As a provider, if I had a patient in your situation I would be willing to educate your command and/or write you a profile stating it is medically necessary during your transition. These are just a couple of outside-the-box examples your med/BH providers might help you with.

AR 670-1, Section 2-10(d) states "Soldiers will comply with all standards for uniforms and grooming for the Soldier’s gender. The Army recognizes a Soldier’s gender by the Soldier’s gender marker in the Defense Enrollment Eligibility Reporting System" This is where Leadership may get you in regards to not being allowed to wear a bra, especially when coupled with DA PAM 670-1 Section 20-32. In that section it states that females will wear bras with all uniforms. It does not state that men can wear them (it also doesn't state that males can't wear them). What I mean is that since you stated you are going thru the Transition, and last I understood the policy (unless it has changed) that you will wear uniforms in accordance with your gender that is marked in DEERS. And that you cannot change your gender in DEERS until such time as the transition process is completed.

Again, this is how I can see potential friction with your Command in regards to wanting/needing to wear a bra. Personally, I could give a rats ass if you wear one. But that's just me. I recommend you seek out legal guidance from JAG so that you can have a clear cut understanding for this.