A quick update here for what I ended up doing.
I bought four “mini” Lenovo business PCs - they are about the cross-section of a shoebox, but only an inch or so wide. I found a seller on eBay UK who really knows their CPU/MB specs, and I could customise quite a bit. Each unit cost about £110, if memory serves: 16GB RAM, 120GB SSD, no WiFi/BT, Gigabit Ethernet for each.
In the end I only used three, but I might still add the fourth. I put Ubuntu Server onto each of them, then Canonical’s MicroK8S, and then joined them in a cluster. I added a cron script on all of them that puts each to sleep at 1am. I then added an always-on Raspberry Pi to act as an external bastion and as a wake-up server. The wake script just uses Wake-On-Lan to unsnooze the nodes, and they wake up within five seconds or so.
The total current consumption when alive (but with no effective workloads) is around 70W, the consumption once sleeping is 8W, most of which is the Pi.