It is definitely not from mosquitos in this weather. Ticks go dormant with the first frost - those Lyme disease bites usually have a bulls-eye pattern. The white raised area is the bite, and the larger reddish area is either the outlying area affected or possibly due to her rubbing/scratching it, or both. I get bites like that from the May flies (the Massachusetts equivalent of the Maine/NH black flies - a hard bump and a larger area where the reaction spreads. In my case, the reddish area can be 4 inches across, and it's hot to the touch. But again, it's not the season for these flies.
Flea bites usually don't occur when the cat is still in the house - those occur when people take the dog/cat to the kennel before a vacation, and then flea eggs hatch in the house. With no animal to feast on, the fleas bite humans. However, fleas don't fly, they jump, so bites (at least on adults) tend to be on the legs - with a child, they could be lower, especially if the child sat on the floor. But since your cat is in the house, I doubt that's it. And your child would have a whole bunch of bites, and you would have them too. Fleas are very tough to see - the best thing is to check on a place like your cat's stomach where there is very little fur - if you see something scurrying along, that's it.
Bed bugs - I would think there would be a ton of bites.
My guess is a stray spider - now and then someone is susceptible to them. I'm usually the one in my family. Sometimes they will do a few bites - do you find them in a row or in the same area? I've never actually seen the culprit in all these years - I just see the results occasionally. Often it's in a crease in the skin or along the waist where the elastic of my pajamas is, or along my bra line. Not sure why - I imagine something gets in there and likes the close quarters, then has a little feast. But that's not scientific, just my supposition.
Poison ivy would be a rash without a raised center as you describe, or it would be blisters, or both. It can occur year-round as the oils stay on the stems of the plants/vines long after the leaves have died back. That can come in on the cat too. But this doesn't sound like poison ivy. It also doesn't sound like anything from the tree.
I really wouldn't worry about it. The fact that she gets them in the living room is the only thing that stands out as interesting or worthy of more investigation. Is she always on the same sofa or always on the floor? That would sort of pinpoint an area. All I can think of is to do a thorough vacuuming of the upholstery and the floor but otherwise just keep an eye on it. I don't see any imperative to go in for extensive allergy testing. If it's an insect/spider, which I suspect it is, it's not going to show up on any test. And what are you going to do about it?
You can use OTC anti-itch products, but be aware that the rubbing on of the cream can intensify the itch. There is a product called Domeboro which is a powder you mix in water and apply as a compress with a washcloth - just place it over the bites but don't rub. It's particularly good at drawing out the moisture of a poison ivy blister but it might give her some relief. Otherwise, maybe there's something you can dab on with a cotton ball. I don't find calamine lotion to be particularly effective but it's an option to look at. Otherwise, if she'll let you, put some ice on it to reduce the swelling and quell the itch.