Can my father declare on his taxes the income from renting out 2 rooms in a house (in which we both live), if he transfers the house to me? Or will I be required to report this income?

My widowed elderly father and I are currently living together in his house. He isn't doing too well healthwise.

Given our financial situation due to coronavirus and some other factors, we both agreed that we need to rent out two rooms. We will both continue to live in the house, alongside the 2 new room renters. He is the sole owner of the house (only his name is on title), and only his name is on the mortgage loan.

For specific reasons well beyond the scope of this question he would like to transfer the house to me soon, instead of letting it pass through inheritance/his revocable living trust. 

After the transfer, I will be in the unusual situation where my father's name is the only one on the loan, and my name will be the only one on the title. I will not be assuming the loan.

Some background (not sure how much of this is relevant):
-The house is his principal residence. Neither of us owns any other real property. We’re in California.

-We did some Medi-Cal planning, and structured our bank accounts and finances so that he will be eligible. I do not pay rent.

-It is our understanding this house transfer (his principal residence) should be exempt from CA’s current 2.5yr lookback.

-We both understand his cost basis will transfer to me (versus inheriting at FMV on date of his death). Because he received a step-up in cost basis when my mother recently passed away on both her half and his own half (double step-up), this isn’t really an issue. The house value/cost basis probably would not be much higher if we waited until inheritance.

-The Garn St. Germain Act permits transfer of a house, parent to child, without the loan being called. The lender is legally forbidden to do this in these types of transfers.

-Just to reiterate : The house must be transferred soon, and not inherited, unfortunately. This is not something we are doing without proper legal counsel, or are doing on an uniformed whim.

-The rent checks will all be made out to him. He will pay the mortgage loan (which is in his name) with them and keep the rest.

-(Property tax and homeowner’s insurance will be switched to my name. The mortgage payments include the monthly escrow payments for these.)

-All utility bills are in his name. He will collect the renter’s portions of the utility bills, and pay these bills himself electronically through his bank account.

All of this would be fine, except that now we are both concerned that I would have to declare all the rent income that he collects for himself (to keep and to pay the mortgage, which is in his name), on my tax return instead of his. Without going into detail, financially overall, it turns out this would actually be a big negative, and would affect my health insurance coverage (Covered CA), among other things.

I’ve read through some posts describing situations where non-owners of homes claimed rental income as their own, and advice saying this can be done sometimes, but none of them exactly matched our situation.

My question is, even though I will have sole title to the house, will my father be able to declare the rooms' rent income on his tax return? Does the IRS let us decide who declares it in this situation? Or will I be required by law to take this income as my own? Again - the renters will pay him only, and he will deposit the money into his account to keep and to pay the mortgage (which is in his name only).

We both are okay with not taking any deductions associated with renting the rooms if this will permit him to declare all the rent income on his taxes, although it would be nice if some deductions would still be available.

Thank you.