Hardware needs for minimal DVI processing with precise timing requirements
Are microprocessors sufficient for this application?


No, Video links in general and high-definition digital video links like DP/HDMI/DVI especially are really high-rate applications. You can't even process this in real time on a normal PC CPU, from the sheer volume of data alone. It takes dedicated hardware to generate it in a GPU; DP can carry significantly more bits per second than datacenter-grade network links (10GBase-XX ethernet). (note that I don't say a PC can't generate 240 Frames per second, I just say that the resulting rate in raw video is too much for normal PC-style computing. I do multi-100-Megasample/s signal processing.)


  An FPGA board feels overboard, but I'm unsure.


Absolutely not. ASICs and FPGAs are how you deal with video signals.


  Do you have any recommendations? What about if I wanted to modify the color of every pixel slightly in an independent way (increasing brightness, say), to help with calibration?


Get an FPGA board with the necessary interfaces.

In fact, there's already projects that process video, and forward it, for conference recording purposes: See mithro's 33c3 talk:

https://fahrplan.events.ccc.de/congress/2016/Fahrplan/events/8057.html


  I'm trying to avoid implementation complexity if possible.


I think you might be on the wrong project than. DVI/DP/HDMI is not a simple thing, and the timing requirements for any processing done on are not easy to meet without significant FPGA design experience.