Your half brother to whom you are not actually related to by blood, who is not related to your mother and was estranged from her for 20 years, feels that he should 50% of your maternal grandmothers estate on the basis that you mother may or may not have promised it to him.
There is no evidence to support this claim, you have 4 of your own to support and have made an extremely generous offer of 25% on the grounds that he is your last remaining "family" which has been rejected.
Couple of things :
As a non relative he will pay capital gains on whatever he receives. You should seek financial advice about how you could gift a sum in the most tax efficient way.
You should put this money or part of it into trust for your children and ensure it is locked away. Seek financial advice
You should write your own will. Pronto. Seek financial and legal advice.
You don't mention a spouse/partner ex or otherwise. If they have a claim on the money [that you don't want] you should seek financial and legal advice. If you have a spouse who you love and trust, I suggest that you listen to their advice.
"Family" that will remain close to you only with financial inducement are not worth having in your life. I'm very sorry.
In the meantime you can stonewall your half brother with the likely fact that the estate will be in probate for some time, possibly years due to the backlog in the courts. Substantial amounts of tax will have to be paid on it and having considered your original offer [rejected] of 25% to be exceedingly generous and unnecessary you have decided to get financial and legal advice once probate has finished.
For the comedy value - ask him if you can borrow some money temporarily to sort out all the advice not the grounds that his mother left him well off. Watch him show his true colours.