Put AMD FirePro D300 on external case?
The MacPro 2013 GPUs aren't standard PCIe cards. I don't think anyone's made an adapter that can use them.
The trashcan won't boot without Graphics Card B which also hosts the internal PCIe flash drive. As @joevt said, there's no adapter to use the proprietary FirePro DX00 graphics cards in an enclosure. Your best bet is to pair a standard PCIe Radeon card with a Thunderbolt 3 enclosure. You'd need a software unlock such as Purge-Wrangler to use the eGPU with the 2013 Mac Pro. Here's an example of my 2013 Mac Pro running with Mantiz Venus + Radeon VII eGPU.
I bought the cheapest Mac mini 2018 (3.6 GHz Quad-Core Intel Core i3). It has faster single and multi thread scores in Geek Bench 5 than my Mac Pro 2008 (2 x 3.2 GHz Quad-Core Intel Xeon). Check the benchmarks to see how well a Mac mini will compare with your Mac Pro 2013.