Define reasonable
Newegg in the US has the Vega Frontier Edition for $999..
If there are things you want me to try with available tools, I'm happy to give it a go on my box. So at least you'd get confirmation that the Vega would solve your problem.
What does your "System Information" gives about the vega in Graphics/Displays and in the PCI section?
While at it, any chances you could export the SSDT/DSDT and the IORegistryExplorer output?
I didn't do so while I had mine at hand unfortunately...
If you can, I'd love for you to try the following:
1). Download and install a free copy of Davinci Resolve 14. Enter
any credentials when prompted. It will start downloading the file immediately.
2). Open up Quick Time Player and do a 30 second screen recording. You'll have to time it yourself, as there is no on screen counter for screen recording. Remember to "save" the file and give it any name you want.
3). Open Davinci, create a new project, go to the "Edit" window by clicking the "Edit" icon at the bottom of the Davinci Resolve window. Drag your screen captured footage directly onto the timeline, starting from the very beginning of the timeline.
4). Go to the "Preferences" dropped down from "Davinci Resolve" task bar item in the upper left corner. Make sure you are set to OpenCL and that your GPU is manually selected.
5). Go to the "Deliver" window, which is the indicated by Rocket Ship icon on the bottom of the Davinci Resolve window, and change the render setting (in the "Render Settings" box on the upper left side) to use the YouTube preset for 2160p.
6). At the bottom of the "Render Settings" box, click "Add to Render Queue." Choose the destination for where the file is going to be outputted.
7). Once it is in the queue (right side of Davinci Resolve Window), push "Start Render" to begin the h.264 encoding. Once it is done, report pack with the time it took for the render.
On my iMac Pro, exporting a screen recording to h.264 4K took 24 seconds to complete.
If you finish your test within a margin of a few seconds, give or take, then I'd consider that normal. If it takes you anywhere north of 1 minute, then I think there is an issue.