Yeah I just tried firing Doom 2016 up again, been a while since I played it. You're right, Fraps doesn't work at all with it (at least with Vulkan), and although the built in FPS counter can measure Min, Ave, and Max FPS when set to High, the Min, Ave, Max results shown in green text, are way lower than the actual.
I can't see giving up the optimization of Vulkan if it's only broken on that API. If you're just wanting to monitor while choosing best settings, I suggest if you need to turning down Shadow Quality, Player Self Shadow, Decals, and Lighting, in that order.
It can make a significant difference on FPS just going from Ultra to High on those settings, and there's marginal visual difference. On a 7970 I can run it with everything maxed except shadows at High and get 55-60 FPS with adaptive Vsync on, but with player shadow, decals and lights also set to High, I average 70-100 FPS.
Here's a handy guide that shows the visual difference of all settings and their performance impact.
https://www.tweaktown.com/guides/7757/doom-graphics-performance-tweak-guide/index.html