Is the Imperial Hotel in Vung Tau guest friendly?

All hoteliers make money with guest friendly facility. If they stop it their occupancy will gone down

So does that mean The Imperial hotel will allow me to have an overnight guest?

I recall that at the Rex Hotel 20 years ago, daytime visitors had to leave their government ID's at the front desk.  I expect that has changed.  I also have read on this forum that the rules against cohabitation with foreigners also have been formally changed.  I expect that the hotel may just be being conservative, in particularly trying to avoid a reputation as a place frequented by prostitutes.  I don't mean anything about your visitors; just speculating about the hotel management's perspective.

You might also check on the rate structure.  Perhaps an additional person in the room changes the rates which would of course give them a legitimate cause for concern.

Most government run hotels wont allow overnight guests and some private hotels will charge for an extra room. Instead of asking on here, contact the hotel via email and get their policy first hand.

Why not just register as one room, two persons? If you haven't "arranged" your date yet, just say she is arriving later.
My Vietnamese wife and I (Caucasian) are always asked to leave our IDs at the desk as is normal for hotels. She is just the woman who is with me, our relationship is never questioned.

yes

Contact them and ask. Stayed there not long ago and it's a pretty upmarket place. You will probably need to register anyone you take in and possibly "rent" a second room for your guest