The childs father should be doing everything HE can to get the money, sell his car, get a better or second job, everything HE can.
Much easier for him to nag and sulk because the OP won't be a total mug and cough up.
OP let him sell you his half of the house for cash, and pay you rent going forward if he is so keen.
Blended families often keep finances mostly separate.
My friend has two older step children who were privately educated by her husband and his ex.
They are in their 40's and received a modest inheritance from her side of the family.
She shared their living costs with her husband when they lived at home as teenagers.
My friend had a daughter with her husband who is now 25 and she has recently come into a substantial inheritance from my friends later father. Her daughter was not privately educated as they simply couldn't afford it with university for the older two etc.
They were all very surprised by the amount her daughter inherited, and that my friend had asked her father privately to skip her and give it directly to her daughter, who will now buy a lovely home for cash.
Her husband is very upset at her "trickery" to avoid his children getting a share.
She has put him very straight that her father owed his older children nothing.
He wouldn't let it go, got quite nasty about it so she packed a bag and has gone to stay with her widowed cousin, and told him she wants a divorce.
He is 70 and haquickly y backtracked but after a couple of weeks away from him and his controlling ways she has said no.
He is no doubt missing her cooking and being looked after well as he is 10 years older than her.
She is loving being away from him and has said she really is done and this row about the inheritance is a blessing.
She likes her step child but does not feel they are due any inheritance from her or her family.
They have inheritance from their parents and that is enough.
Her priority is her child.
I think that is perfectly reasonable.