I'm going to answer what I would do in this situation. If I had a partner who was allergic to my dogs, and he wanted to move in and requested that I rehome my dogs so he can do that, I'd break up with him.
It's not that his request is wholly unreasonable. If someone has a severe allergy that cannot be managed, it's reasonable to expect that the cause of the allergy be removed from living space.
It's just it's not something I'd ever be prepared to do. I do not care one inch that I'm happy to prioritise my dogs over humans because I absolutely am. I made a lifetime commitment to them for the duration of their lives, and I will see that through.
What have you both tried before deciding to rehome your dog? Has she been to a doctor or pharmacist for advice about managing her allergy? I understand this isn't always feasible or possible in some cases-but I think it's worth a try before moving straight to rehoming.
Have you tried having a good deep clean of your home? Regular grooms of your dog using deshedding shampoo? Limiting the places in your home where your dog can go-i.e. the bedroom is a no-go area.
It seems that all avenues to resolving this have not been explored. You're digging in your heels because, deep down, you don't want to do it. That's understandable. She's pushing for a resolution because she doesn't have the same emotional attachment to the dog as you do. It's also, in a way, understandable.
I'm not going to say that she's a villain in this piece-if she's severely allergic and unable to manage her allergy through treatments etc, her position is logical and reasonable. It's possible that because she thinks the dog will get rehomed with a family member or friend that it's somehow less of a sacrifice on your part as you would still be allowed to see it.
If anything, I think you sound quite immature and naive. You are perhaps young and as you haven't been together that long-I'd suggest taking a deep breath and pause. If you are, indeed, partners for life, what is the hurry to move in together? If she doesn't have a lease that's due to expire, gets on with the relatives she's staying with, where's the fire? Why June?
I'm sorry, OP, I don't see this ending well. Even aside from the issues with the dog and allergies, it seems like you're quite nervous/scared of communicating with her.
It's clear from your hesitancy to act that you don't really want to do it. If you did, you'd have asked around friends and family by now. Yet, you don't want to say anything to her about this in case she views this as you "going back on your word." I'm sorry-but any partner that has empathy would be able to understand the difficult position that you are in.
I'm trying to think of what it is that you want from MN, OP, because many have given advice about things you could both try that don't involve rehoming the dog-but it seems you may want some handhold that lets you off the hook here. Please don't ever think of owning a dog or any animal again, if you treat them as disposable depending on the status of your love life.