You can't shut the dogs in a room for an hour or so? Is there someone who could take them for a nice long walk? Despite the looks they may give you, they won't actually die without you! Spaniels really do excel at emotional blackmail, don't they. I would seriously take an evening of canine sulking over weeks or even months of putting together bits of quilt in that way you're proposing, it honestly sounds like a total nightmare to me.
Funnily enough, I was talking to my mother about using the Miracle Ball Method, which is basically when you lie on one or two softish balls the size of a grapefruit and it helps your back. I did a demo last time my parents were visiting. They had their young Yorkie with them, and she thought it was absolutely fantastic when I lay down on the floor. I was completely at her level! Obviously it was a game! It was somewhat tricky to lie back and relax and do deep breathing with this tiny thing leaping all over me, not to mention that I kept on laughing.
Carpet - is it low or high loft? Apparently big pins work, you poke them in at an angle so that the head of the pin is closer to the quilt edge than the point of the pin, if that makes sense. It's in Jinny Beyer somewhere, that queen of hand-quilters. If the carpet is fluffy, you have problems (in which case see if there's somewhere else you could baste, a friend's house or something), but if it's completely flat, you should be able to do the pin-basting without pinning the quilt to the carpet.