best choice for the facia boards or rim boards.
Fascias are trim, just hanging out there.
Rim joists are structural, occur at the wall, and take compression loads.
Assuming fascias…..
LVL etc will be overkill…..not needed for structural reasons.
I would just use 5/4 X 10 or 12….don’t know if you used 9 5/8 or 11 5/8 TGI.
Or 1X trim over a 2X sub-fascia.
Assuming rim joists…..
LVL.