"But he’s renting so right now there is no inheritance that way."
But what happens after he has been living with you, and contributing to the mortgage and family costs for the next 30 years?
I also assume that his income is being taken into account when the bank is working out if you can afford the mortgage?
Are you really going to disregard all of that?
As so many others have said, a fair split would be to get a Deed of Trust that says that you are entitled to a certain percentage of the house based on how much deposit you put down.
Eg, if you're buying a house for £500k and you're putting down a £100k deposit (which is 20%) then you get back 20% of the value of the house if/when it is ever sold, and then the remaining value is split depending on how you each paid the mortgage and contributed to the household expenses.
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"My fear is that he’d die, and his ex would sue me for lost CMS or something"
Oh please, you cannot be serious about this! I refuse to believe that you could think something as incredibly wrong as this.
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"I’d have to give up my child’s home whilst in mourning to give a payout to his kids."
No, you wouldn't. This is why people go to a solicitor to get their will written. A properly written will, will allow you to live in the property (and even sell it and buy another property) for as long as you want, until you die.
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"I don’t see why they have to be given our home."
Wow, I'm really suprised that you're referring to it as "our" home. In reality, you you think of it as "my" home and you are objecting to to him sharing your life.
That's totally up to you, if you don't want him to contribute to and share in your life financially, then that is totally your perogative.
However, you can't then complain if he chooses not to contribute to the mortgage or your family expenses beyond what would be proportionate and then prioritise all of his children, not just the one he shares with you.