You have any number of choices - all unpalatable.
1. Refuse to close on the specific date. You run the risk of losing your purchaser. If you have a queue of people wanting to buy your house (unlikely), you just sell to someone more agreeable.
2. Close on the specific date, have the money in your pocket and be ready to move on. The risks are that you go to enormous trouble to prepare to close but it does not happen and you have stressed yourself out for nothing.
3. Talk to your solicitor. Go through all the pros and cons and make a decision that gives you the best result.
I rather suspect that you are on a tightrope and that, if you do not at least try to close on the specific date, and bearing in mind that your contract is probably so wide open with get out clauses for your purchaser ( and that's the way it is right now - vendors have very little leverage) that it might all work out and , equally, it might not.
Its unfortunate but market forces dictate.
It is all entirely legal. Its all about people agreeing to terms in a contract. You do not have to do anything you do not want to do BUT that may mean you end up not selling your property and remaining stuck where you are.
Talk to your solicitor and work it out from there.
mf