I worked at a large sign shop where the majority of the signs we did were back-lit.
About a 1/3 were painted, 1/3 were cut vinyl & 1/3 were printed, depending on the job.
1st off: Your face, I definitely would have it painted or use cut vinyl, it would be easier and cheaper.
2nd: The only true way to make your print look the same during the day and night is to print
it twice, apply the 1st layer, then apply a white diffuser , and then apply the second print.
The 2nd print is typically printed at 1/2 the density of the 1st print (assuming you're applying 2nd surface).
When we started printing white, our printer would print color - white - color all in the same
pass, so we could do a complete print on one piece of clear.
IMHO doing a double strike print for a back lit, might be okay for mom-and-pops sign
shops, but if you plan to do high end and/or corporate back lit sign work, you should do it properly.