Hi Mick,
I'd like to get a little better idea of what you're doing. First off, you mention "refurbishing" the display. What exactly do you mean by that? Are you simply replacing the screen and moving the earpiece speaker assembly to the new screen? Or are you getting deeper into it and swapping the display IC so as to eliminate the genuine part warning?
Most commonly when there's a Face ID failure following a screen replacement, it's generally due to damage to a flex cable on either the dot projector, the IR camera or the front-facing camera. If it's the front camera, that's actually replaceable as it's not paired to the logic board, but it'll take some Dremel work on the metal housing. Otherwise, if there's damage to one of the other two flex cables, they may be repairable, but you have to keep the original parts in order to preserve Face ID.
You might want to check into methods of verifying the operation of either the IR camera or the dot projector to see if you can pin down where the issue is coming from, as it'll most likely be one of those two things giving you trouble.
Oh yeah, check the peepholes in the display for any of those blue protective liners that may have been overlooked; if there's something in there it'll mess up Face ID.
There was a short period of time in which Apple was disabling Face ID if it detected that the screen wasn't original, but they later relented on that, so make sure and update to the latest iOS release to make sure that's not the issue.
That's all I can think of at the moment; good luck with it and let us know what you find!