The darkening of the screen is a relatively simple issue, but depending on their engine pipeline might be tricky to solve.
Modern games use a range of light far beyond what a display can show - it's entirely linear, and then at the end of the process this is tonemapped where all the values are brought down into the visible range, setting the overall contrast and also the peak white point for HDR.
The issue comes from motion blur being calculated after the tonemapped HDR image is encoded - so when you take a pixel thats got a value of 1.2 vs the original un-tonemapped 20 (for the sake of simplicity, 1=white here) and blur it against something dark, it finds a new value somewhere in between, dimming the white point.
If motion blur is calculated before tonemapping, the ultra bright points of the linear image burn through any blurring and retain their true value. then you tonemap, and everything looks as it should.
Games that fake HDR dont have this issue, because they pop out the higher dynamic range as a post process anyway.
If they really cant reverse the order of the process, what they might have to do is spit out a frame buffer channel of the original peaks above 1 during tonemapping, then after the motion blur is done use that as a guide to do post-process 'faked' HDR (from an accurate reference) to pop the highlights back out to the correct value - which should leave them stable.