I'd say they are likely to be full but you may be able to call and find out. There is another lake by there with at-large camping, called Scott Reservoir, over off of Porter Mtn. Rd. There might be a restroom but nothing else there. But I expect it will be full, also. Are you looking for a lake to camp next to with kids? We did enjoy Show Low Lake and Fool's Hollow when our kids were little. I think with Fool's Hollow you can call and see about reservations. I am not sure about Show Low lake in that regards, it is a city park, (I think,) whereas Fool's Hollow is a state park. (The state parks people changed its name, without anyone's permission, to "Fool" Hollow, although I noticed the new road signs say Fools Hollow.) We liked Show Low lake back then because there was a campsite right by the lake, which we were lucky to grab.
Big Lake has tons of campsites, have you checked that reservations website to see if there are openings?
Mexican Hay Lake allows at-large camping, but they no longer stock it with fish because it dries up in the dry years. You could camp there, right by the lake, and drive up to Crescent or Big Lake for the fishing, though. To get to the campsite area by that lake, you have to go through a gate. You'll see it by the highway. Usually not many people camp there--sometimes no one is there at all on a beautiful summer weekend. And there is a good hiking trail by there if you can find it. It is the upper end of the South Fork Trail #95. Let me know, I'll give you directions to find it.
Hulsey Lake by Escudilla Mtn. has no campsites right next to it, but there is at-large camping at Terry Flat and a regular campground at Alpine Divide, right next to the highway. That lake used to be the little unknown fishing spot, but is now extremely popular. They had to expand the parking area.
And then there is Carnero Lake by Green's Peak, off FR 117. It has at-large camping right next to it, but only two or three sites. Could be good, or could be a party, hard to say. Very nice place when you have it to yourself, though. They do stock it but you would have to call G&F in Lakeside to find out when they last stocked it. Lots of at-large camping all around that area, though, but right next to the lake there isn't a lot of room.
Luna Lake in Alpine has a campground, but it is not right next to the lake. There is at-large camping along the road that goes behind Escudilla, not far from the lake. ATVers love that area, though, so might be noisy.
Hoyer Campground and Benny Creek campground in Greer are both very close to the three Greer Lakes, but you can't camp at-large by those lakes. Not sure if Hoyer has re-opened yet. They have been busy re-doing it.
Ok, I'm done. I don't know about the lakes on the reservation at all.
There is a point of no return unremarked at the time in most lives. Graham Greene The Comedians
Steal your face right off your head!