Thank you to everyone for all of your advice! Loki stayed in his crate last night in our room, and I ended up sleeping at the foot of the bed so he could see me, and it really made a difference. Much less crying for him, much better sleep for us. It's not the most convenient thing to carry the crate up and down our stairs, but hopefully he'll graduate to his own bed soon.
We think he may also be cold, making the crying worse than just being lonely. His crate is loaded with fleece, but it threw my electric blanket in his crate this afternoon just to test it, and he is very obviously waaaaaaay happier in it now. He still prefers to sleep in my lap, but he stays in his crate when I lay him in it as he's sleeping. Definitely a temperature issue. We left him alone in the crate for 30 minutes today, and he cried much less, even stopping briefly by the time we walked back in (we left a video call on my laptop so we could watch him while we were gone.)
He's still not messing in his crate, which am supremely thankful for. No accidents today, either. Now, to just work on the biting…
Thank you all for your support! Here are a few photos to show you all what we've done. You'll notice that I have part of the blanket under the door, so he can't chew the cord. Puppy life is so rough