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You could just be enigmatic and say 'that won't work for us'. Or invent a reason, like oops we a now in a chain and have deadlines. Or that you incur some kind of fee if you don't move. See what they come back with. It's the slightly less nuclear option. That way you can at least try the 'OK but consequences' above if they remain stubborn on the dates.
If they get a bit shitty about it, then absolute worst you can mumble some kinda of half agreement, then drag your feet on exchange as long as possible and generally punish them for being dicks.
They can't ask for something without giving something, and they didn't give you this information when you made your offer, so it's fair to negotiate.
They are coming from the "well, we are saying yes to 30k under asking, so for us to say yes then we want to stay until March" angle. Which is kind of bullshit, and I do want to say jog on, pay us rent from completion until you move. But at the same time this is the house we want by a long way and are okay with bending over a little bit so we get it. Just want to make sure we don't bend too much...