Not to sound trite, but according to Law of War the US MIL may very well be operating under a scenario of beligerant government. That is, recognizing the government civilian authority has been taken over, and undergoing steps to correct it.
If something big like a war were to come to a head, then they would have to make it known that they did not recognize Biden as the authority.
Until then, they could easily keep public awareness and internal operations hidden from 'the enemy', meaning Biden and his illegitimate puppet government. In other words, cooperate only on an outward basis for appearance-sake. It is why we would see things like AF-1 not being used as a call sign. Biden may SEEM to be in control, but what would really happen if US forces were called on in a serious way? By serious I do not count air strikes on Syria.