I bought a Shield TV and the cheapest (non-free) monthly Geforce Now subscription, paired my DS4 controller to the Shield TV and it's pretty much worked. The DS4 is read as an Xbox controller so a couple of the button mappings needed reworking (and there's no rumble, sadly), but it's generally been OK - can do everything with the controller that I could do on the PS4. It's more of a faff to launch the application and takes a minute longer to load (planet generation every time), but I can still lounge around on the sofa while playing, and an unexpected benefit is that it's much quieter - the PS4 fan could get quite noisy, whereas the Shield TV is silent.
I'm still uneasy about having to do it at all, but it's wayyyyyyyyyy cheaper than putting together a rig. I find it ironic that it was uneconomic for Frontier to continue development on the console, but as a result I've now splashed a bunch of cash to other companies in order to continue playing their game...
I know I could have stuck to legacy/PS4, but after the 6 days notice they gave for console transfers I figured they could be equally brusque whenever they decide to pull that.