Lizards are slippery little fellas, I had a panic attack a few days ago when my tegu escaped somewhere in my room, we found her asleep in the closet after tearing my room apart. I have no idea why she chose the coldest place in my room to sleep but I suppose when you want to sleep anything works well.. tegus and their winter cool downs are so annoying.
Pretty sure if I die it's going to be caused by her with how much stress she gives me.
She's not full grown by any means but is 22 inches long and as big as a beardie in girth almost, she can probably get in more places than a beardie can for now (she still fits in paper towel tubes and has the girth of about 6 inches at the widest portion of her body), i'd look in that spot that isn't sealed, thats most likely where he escaped.
As for your dogs probably put them in a kennel of sorts while you look, unless its warm outside I wouldn't put them outside.
Maybe setup a basking spot if you can and see if that lures him out.