Here’s a typical position during the work day when we have the dog cage around the couch. Me, tea on the lap desk (computer’s to the left—wisely I bought a big desk) and my leg stretched out so Trixie feels she’s sitting sort-of in my lap. It’s awkward, but she often prefers to be inside the legspace. And no, the tea/computer positioning is not ideal but I don’t have much alternative where to place the mug when I’m on the couch. I’m very careful and so far I’ve been very luckytoo.
Hook a cup holder through the cage bars, maybe?
They’re not very stable. I do have a kind of mini-table on the couch arm that I use sometimes, but I’ve knocked it off a few times too.