Uploading in case someone finds my build useful. The Sapphire Nitro fits in the H200 with 2-3mm spare by the fans, and maybe 5mm on the underside.
Temperatures while gaming with 100% GPU utilisation sit around 82C (while no frame rate cap applied). The GPU fans are actually relatively quiet, and my case fans end up being louder while under load. I think these temps are fairly normal for a 5700 XT.
Tested a range of games and find that my older CPU still seems to do a decent job of feeding the GPU (CPU utilisation around 40-80% depending on the game). I game on a 75Hz LG 2560x1080 ultrawide monitor, but if you're trying to run super-high refresh rates it might be the weak-link however. I will probably upgrade in another year or so.