I'm not clear whether this is a proposal to upgrade your living circumstances into a much more expensive house or a proposal to buy a house to rent/flip? Or something in between: move in for a while, add value, and then sell and move on somewhere else? It changes the considerations a bit depending on which you are looking at.
Apart from that, I would have thought that someone's main reason to sell at an undervalue is to get a quick and troublefree transaction. Your proposals to lock the seller into a potential delay of up to 2 years would seem to negate those reasons for them to sell to you.
Looking at the money side of things -
1. Have you contacted an estate agent about selling your own home? If this opportunity to buy really is such a bargain, then you can probably sell your own place at a slight discount in return for a buyer ready to proceed quickly: that should take away a lot of the strain of a simultaneous sell and buy.
2. You are going to have to borrow: either about 175k if you sell your current house and 455k if you don't. Do you have a financing offer in place, and what's the interest rate?
3. A house with a kitchen and decor that need updating will be harder to rent out and will pull in less rent. If you are going the rental route you should probably factor in the time and cost of a quick cosmetic overhaul. A rental income of 4k a month should cover your borrowing costs even at the higher amount if you can get a decent interest rate - which makes me think that you shouldn't need to faff around with a delay in the purchase terms, just buy the place outright.
4. You say you've a couple of big surgeries coming up. That means that you won't be doing much if any of the work on the house yourself but will need to hire it out. That probably adds cost to the renovations: the house may still be a bargain but a bit less of one to you than to someone who can do the work themselves. But if you are FIREd it doesn't otherwise change the financial calculations: as long as the house at least covers its costs as a rental you and your family can weather any delay caused by illness/disability etc. until you can sell on and make the capital gain.