Should I factor in the $6500 into the cost basis of the property specifically for my Schedule E entry for Bedroom 3 (master)?
yes, you can add that to the cost basis for bedroom 3/rest of the house, and the in service date would be Dec of 2022.
You don't have to list it as a separate asset for bedroom 3. Since the work was done 10-11 months before it was placed in service, the rest of the house (with bedroom 3) and the improvements were placed "in service" as a rental all on the same date. So you can just add the cost to the bedroom 3/rest of house cost basis.
The cost basis for everything should add up to your total cost basis in the property, which includes the $6,500 for the master bath.
What about the other bedrooms?
What about them? You've accounted for your total cost basis in the property already. There's nothing left to add to the cost basis of anything.
Can I expense a portion of the cost of this repair/improvement(?) as part of the De Minimis safe harbor
No, because the cost of the improvement exceeds $2,500. Therefore, the safe harbor election is not an option.
Your real issue is going to be the 2023 taxes and getting the depreciation correct for 2023. That's asuming the property remains 100% rental for the entire tax year of 2023. It's going to require a lot of manual math on your part. But don't concern yourself with that right now. We can deal with that next year when you file your 2023 tax return. I would just suggest you bookmark this thread for reference, so you don't have to repeat everything when the time comes.
