I was hoping someone with more definitive knowledge would answer this one, but it's been a day and nobody has yet, so I'll offer what I know that's related to but not quite the answer to your question, and what I guess is the answer. (Short version: I'm guessing the answer is no.)

But first off, what I know:
The only part that comes with the 30 days of subscription time included is the base game, ARR. Expansions don't add more time to that, though adding ARR for another platform does (e.g. adding ARR for PC to an account that already has ARR for PS4 or vice versa).

The Complete Edition is essentially a bundle combining the base game and expansions into one. Its intended audience is new players who don't have either yet and buy the Complete Edition in order to get everything at once. When used that way, it provides 30 days of subscription time because it includes ARR, which provides that 30 days.

In fact, when the Complete Edition was first introduced, it wasn't even possible to add it to an account that already had ARR (unless for a different platform) because when the system attempted to add the ARR piece and found it already there, it failed saying the code was redundant. After complaints from some customers trying to use it to just add the remaining expansions, SE relaxed that restriction so that, so long as at least some parts of what the bundle adds are new to the account, it can be added, even if other parts are redundant.

And from here on, I'm into the realm of guesswork:
I think the way it now works, entering a Complete Edition on an account like yours would:
  • attempt to add ARR, but find it already there, so it can't
  • attempt to add HW, but find it already there, so it can't
  • add SB
  • add ShB
So the final result is that all it would really be doing is adding the remaining two expansions, neither of which come with subscription time. If I'm right about that, it ends up being just like adding Shadow Bringers by itself.