OP you work full time, have a five year old with school runs, homework, clubs and play dates as well as a 9 month old baby who doesn’t sleep through, nursery runs and eventually all the additional activities. So after picking up both DC from presumably 2 different locations, cooking tea and bath and bed you must be shattered. You are doing all this on your own for 3 out of every 4 weeks! And your DH expects you to take on the caring/housekeeping responsibilities for his DF on top as well as losing your own privacy and limited freedom. It really sounds like you’ve got more than enough to deal with already OP.
I dare say he wants DMIL to die without the worry over how her widowed husband will cope. It’s too late now but basic life skills of cooking, cleaning and laundry should have been learnt a long time ago. However it’s not too late to learn.
I haven’t read any information regarding DH’s work arrangements. For those 3 weeks is he moving around, abroad or in one place in UK. If in one place could DFIL stay with DH there or buy a place there and DH stay with him for those 3 weeks?
I wouldn’t agree to a trial, it may be better for DFIL to come to yours for a week after the funeral while DH is off, you go to work and don’t make any special provisions. After that week DH could travel back with his dad and start teaching him some basics and see what there is locally to get his dad out a bit. DH needs to look into how he might need to change his work and lifestyle if his dad were to move close by. He hasn’t done this to lighten his own wife’s load and spend more time with her and his DC. I’m therefore not sure he’d be willing to do it for his presumably recently retired, physically and financially stable Dad.
So ultimately no to moving in, no to trial moving in, no to adding a grandad annexe or having him move into your street, no to him handing over money. Discuss increasing FIL’s independence, changing DH’s lifestyle, buying a small low maintenance flat or bungalow and learning to keep house or him paying someone to or him moving into a retirement village. The last suggestion might offer the best of everything for all of you. The one local to me has a gym, bar and social club, large lounge, library, cafe and bistro, hairdressers. They also have occasional evening entertainment and organised day trips and even rooms that can be rented out for visiting relatives. It’s certainly not a nursing home.
Whatever happens I can’t see FIL moving in or too close to you being a positive move. He could have 30 years left, he needs to live them and make them count. This will be a new chapter in his life it doesn’t need to ruin his son’s marriage.