I would NEVER advocate for abandoning an animal, NEVER, but in this situation, I think finding the cat and confining her and bringing her with you might actually be the least kind thing
You said that the cat has not been back to your house in 2 years? That is a very long time. Even when it's raining, even when it's cold?
I think it's very likely that someone else has been feeding her, giving her shelter, that she's got a second or maybe even third house. This is relatively common with outdoor or indoor outdoor cats.
We had a cat come by, I noticed he was around ridiculously early in the morning, I noticed he was out super late at night, and all the hours in between, that he seemed friendly not feral, and seem to be well socialized, but was always hanging around our place. I've known of quite a few cats who the owners moved away and didn't bring the cats with them, and I thought maybe that's what was happening, so I provided this cat with food and water and attention, a safe outdoor place to be... But we didn't take the final step of capturing him, bringing him to the vets etc.
I ended up finding out that he actually does have a home!
I put a collar on the cat, a breakaway reflective one cuz I figured if he's going to be out during all hours, it would help if he could be more visible to cars, and he came back to my place with a note attached to the collar that said that he had a home, it gave the address and a phone number. So I phoned them, turns out they're at the other end of the block, a nice young couple who moved in about a year ago, they were trying to keep the cat indoors for his safety but he really insisted on going out, so they would let him out during the day while they both worked, but the cat just didn't want to come back / go inside. Occasionally yes, he'd show up at the back door and they'd let him in and he'd stay for a while but he wanted to be away more than he wanted to be home. (I think they said that they also had recently adopted a dog, if I recall correctly and that could be why the cat didn't want to spend as much time at home)
Anyway, so I did feel better knowing that this cat did have a home and the people did care about him, the cat just didn't want to be there. There was no evidence that he was being abused or mistreated in any way, and we ended up sharing the cat. Not in any formal agreement, just that they had my contact and they knew where I was, and I had their contact and I knew where they were.
Technically it was their cat, so any veterinary things that needed doing was on them, but because I worked from home, I had more time and attention to give the cat.
He would often follow me for a couple of blocks when I walked the dog, he would keep me company when I worked outside, and one time he came to me meowing and meowing and meowing and it turns out that had been sprayed by a skunk, and I thought oh dear that's got to be awfully uncomfortable for him, so I washed him...(Enzymatic cleaner) And gave him some care, even though he wasn't my cat, that's just the way it went.
And the purpose of this big old story was that sometimes you can have a animal who you care about and you do want it to stay home but circumstances are what they are and animals once they get outside well, they do what they want, and this cat just didn't want to be home. And possibly your cat even though it started off as being your cat, has found another place to be too. Nothing personal. Just happens.
But, if you have any microchip in the cat or anything like that, you better make sure that's removed from your account. That can actually prevent someone else who wants to take care of the cat from being able to give it medical treatment. If the vet scans the cat finds a microchip and sees it's got somebody else's name they aren't able to treat the cat without consent.
Also, this may not be possible, but if you try to figure out who's been giving that cat a home? I don't know if it's possible for you to print a few flyers up and put them in areas that might have the best chance of being seen or if your community has any type of neighborhood Facebook page or anything like that? Basically just tell your story. It's very possible that someone out there has been taking care of the cat and they would love to know what's going on.
My only concern is that if you manage to find where the cat is staying during the day, kidnap her (I know she's legally yours if you have any type of ownership documents but she hasn't been by in 2 years) and take her to a brand new location, that could be quite traumatic for the cat. The cat might even try to find her way back to what she considers to be her home which of course is quite dangerous cuz she might have to travel a long distance through completely unknown territory.
That would be my main concern. That and if somebody else had been taken care of the cat and all of a sudden the cat disappears, it's going to be very awful for those people because they're going to think something terrible has happened to the cat like it got hit by a car or something.
That's all I can advise, hope it helps.
It's very caring of you that even though you haven't seen the cat in 2 years at your home, that you still are concerned for its welfare even in the middle of a move and I'm sure you've got all these other things to think about. That's very kind of you. I hope it all works out.