I'd be hesitant to put the proceeds from the sale of both houses into a joint new one, because your FIL may need to pay for a nursing home or medical expenses.
I think you should keep the sale proceeds separate. You & your wife buy the new house in your names only. Your FIL can pay you room & board for as long as he's living there, which reduces what he will eventually leave to your wife. If you need part of his money to qualify or avoid PMI, then this could be part of his lifetime gift tax exclusion and/or a loan.
My MIL sold her house to move into an independent living apartment in a senior community that also has assisted living if she needs it. This has many advantages: lots of social life (she had become isolated in her house), safe environment (her house was a multi-story danger to her), regular meals (she had been neglecting to eat), & easy access to services (she no longer could safely drive). She had to pay an upfront price that will mostly be refunded to her or her heirs when she leaves, plus a monthly fee that covers most meals, some housekeeping, & other appropriate services. A relative helps manage her remaining money, since dementia is becoming an issue.
You should be careful to select a house that has safe living space for your FIL or can be easily altered for aging issues. These include single story access, no steps, 3' wide doors, grab bars in the bathroom, rimless shower instead of tub, no throw rugs or slippery floors, good lighting.