I have a random idea for how the mixed competition could work, based heavily on how figure skating does it. I know that gymnasts have said before that the competition really long so another day would be difficult. I think that the mixed competition should then be based off an existing competition and the only day that seems fair to me is qualifications when everyone gets to compete.

Each country would get to pick 6 routines on different events before the competition starts and the gymnasts ranking on the event at the end of qualifications would get translated into a point value. The routines would need to be spread across at least 4 gymnasts, with no one gymnast being able to be picked for more than 1 routine. The routines selected must be divided equally between the men's and women's side which means a country must have qualified at least two men and at least two women to qualify for the mixed event (Sorry Philippines). The point values would be decided by overall ranking against all other gymnasts at the games, not just those participating in the team event since countries would likely pick different events to compete. 

Example from the last Olympics:

Simone (Vault)

Simone (Floor)

Stephen (Pommel Horse)

Brody (High Bar)

Suni (Bars)

Asher (Rings)

I'm sure I've missed things but thought this was a fun thought exercise!