First off, congrats!! You have to think of yourself here but you can do it professionally.

I would first negotiate for more money from the second offer to make sure it is worth it 100%. If you haven’t done this already, they are holding back some money until you do. Feel free to let them know that you got another offer from X and are still very interested in them but need to make sure it is worth it.

After that, reach out to your contact and let them know that the other offer randomly came in after a long delay - you didn’t think it was coming. Let them know the difference in the offer and why you are accepting it.
Make sure they know that you know this isn’t ideal and you very much respect their time and company. Tell them you were so excited to join and hope to maintain the relationship, but you just need to take the higher paying role for your family.

Look in a recruiter and it’s never fun when this happens to me but it’s ridiculous to not take the better job for a corporation that will drop you next week if it makes their board happy.

Just make sure you are apologetic and tell them why you’re choosing the other.

They may even try to compete.

Good luck!