Has anyone applied four 200 x 200 mm PEI sheets to a 400 x 400 mm bed?
I haven't tried such a thing, but a speculative answer covering the constraints and expected failure modes may suffice here.
Unless the sheets have squared (rather than rounded) corners, the corners will almost certainly be a problem. In particular you'd end up with a hole right at the middle of the bed.
If the sides aren't entirely square (perpendicular) with each other then you'll have large gaps at one end or another.
As Trish noted in a comment, it's also possible that you may hit problems due to mismatch in thickness between the different sheets, although I'd expect these are fairly well-controlled in the manufacturing process; otherwise I'd expect bed leveling issues even with just a single sheet. If you do hit this kind of issue, it may be possible to avoid it with shimming of some sort.
Assuming you can avoid all of the above potential problems, I would expect what you've proposed to work reasonably well, especially if you're ok with very minor surface defects at the seams.