Need Advice from Cat People, Please

Look at this baby! She (maybe, idk, but I don't see male bits) showed up in my front yard yesterday afternoon for a bit but then left.
Today she's been in the backyard chilling for several hours.
As cute as she is, I can't have cats in my yard. I have a medical needs dog (and I am super allergic to cats unfortunately).
She is friendly (definitely not feral) and seems well fed and healthy. But I've never seen her before and she's making herself a little too comfortable. No collar, no clipped ear indicating she's fixed.

Can anyone offer advice? I want to see if I can grab her and take her to the shelter and see if she's chipped. I *think* I heard that weird "cat in heat" noise yesterday but am not certain. If she's not fixed I'll gladly pay to get her fixed. But she seems like someone's cat so I can't really ethically just catnap it and put it through surgery. (But also a voice in my head says "the owners shouldn't ethically have an unfixed outdoor cat, so maybe it's ok??")

I posted on my neighborhood apps and no luck so far on finding an owner.
My mom suggested I spray it with a hose and this cat was straight up like "thanks for the free shower, can you give me some pets bestie?" so I don't think that's a deterrent (or maybe I'm too chicken to be overly rude to this cat).

She's cute, but she has fleas and I've caught her laying in my dog's outdoor bed. Thankfully my dog is pretty oblivious so far, but the cat caught a lizard and was just messing with it for fun. If there's a dead lizard on the grass and my dog gets it she will get REALLY sick (she has a chronic disease where she can't digest protein properly and we have her on a strict diet and meds with her vet specialist).
So despite my love for cats, this orangsicle is not welcome here.
And for some reason her presence is causing my neighbor's outdoor cat to come into my yard a bunch too. Not sure if that cat is fixed either but at least I recognize that one and it isn't usually a problem. But I don't need TWO cats in my yard.

Any advice is welcome (random opinions aren't preferred, I'm looking for anyone that has experience with rescuing cats or shelter work with cats, etc -- all my experience is with rescuing dogs)
I want to do what's best for the cat while still keeping my special needs pup safe and stress free. I don't mind the occasional cat passing through, but I can't have them loitering or leaving dead animals that will hurt my dog.

Edit: About 10 minutes after posting this, it got in a hissing match with my neighbor's cat (neighbor's cat is mourning the loss of its older brother and probably looking for company) and my neighbor had to come get her cat. I don't want this new cat getting territorial over our yard.

UPDATE: Got a hit from the neighborhood app, the owner is coming over in the morning and hopefully the cat (who is apparently a fixed boy) is hiding under a plant or something and we can get him home! <3

Next day update: the "owner" flaked out and after leading me on all day, finally said it's not their cat, despite the markings being identical to the "missing cat" post. Long story short, they aren't coming. I took him to the shelter, he's not chipped but definitely is a boy and thankfully he's fixed. They're over capacity so they sent him back with me. I still can't have a cat, and he figured out my doggy door this morning after I scolded him for almost killing a hummingbird. Thankfully my neighbor is helping with him, so hopefully he starts getting along with her cat. He's a sweet boy but I hope his owners can wrangle him in soon.