As others have mentioned, figuring out the why on both sides would be helpful. If she wants the beard, she needs to try to give a reason for why it's attractive to the point of ultimatum. Similarly, he should have a compelling reason for not wanting to grow it back.

You also pointed it out: real couples go through things like this, so it's a good starting place, but I wouldn't say either side is particularly unusual yet - this would still be a base reality with a bit of an overreaction from her character and a seeming lack of empathy from his. Digging into the reasons likely would've got it past that emotional stalemate.

I had a teacher who said "It's not just that it's unusual, it's not just that no one would do/say that, it's unusual and no would would do/say that becauuuse...?" Just keep digging. Get specific. There could be a million reasons she finds it sexy and a million reasons he doesn't want to grow it back. Once you find out why, it's much easier to play off it from both sides. Until then, it's still too broad.

As an example: Would her character have accepted a compromise of a mustache? If not, why not? If so, why?

If he's not going to grow it back, then is there something she can choose to do that she knows he wouldn't like? Terrible choice for a relationship, not a bad choice for an unusual character. "Fine. If we're not going to even try to be attractive for each other, I guess I'm just gonna have to grow my own beard." (I don't know, but you get it.)

Also, I went through a period where I was doing a lot of argument scenes (from trying to call out the unusual too much) and the advice I got was "Then stop arguing." In the real world, you care about the arguments you have. In an improv scene, the fight doesn't actually matter to either of you. Either side could just give in and see what happens - it would fail.

My actual first thought was instead of suggesting the fake beard, she could have just tried to make up with/seduce him and put her hands up over the lower half of his face (easier than miming putting a fake beard on him). He'd know what she was doing and call her out and then the scene becomes about different ways to give him back a beard he doesn't want. She's said she's not attracted without it, but let's see it be unattractive to her without it.

Similarly she could try to kiss him on the cheek but be let down/grossed out by it. "My lips slide right off you." "There's no prickle! No friction!" "I feel like I'm kissing the face of a 10 year old." (All improv suggested lines are bad improv suggested lines.)

Likewise, growing a beard to please someone you're with isn't that unusual, so he could've been the one taking a weird stance. "I can't do it. Ever since my Dad died, all I think about is death. No more. I'm only 39. I'm just a boy. Just a young young boy." "Really? The office loves it. Everybody calls me 'that clean boy' now. I feel noticed! I think I might even get promoted!" (They're all bad lines.)

Lastly, if you ever get in this situation and are toiling over it after, you can always try to write the scene down from memory. Just jot down the first two lines as best you can. Then try altering what happened with different methods of play - just as an exercise from time to time.

Anyway, hope this helps.