The bio-wheel, if it was prepared/filled with bacteria at the beginning of the QT, could have gone into the QT at the outset. In this way, the bio-wheel would be going along with the QT treatment.
But, I think if I understand it right, you have begun the QT without the bio-wheel that is still in your MI infected display. Now you want to bring the bio-wheel over to the QT where the treatment is already underway.
If you bring the bio-wheel over to the QT, then in my opinion, you need to start the QT treatment over again. As a hypothetical example: You began your 14-day copper treatment in the QT and you are three days into that treatment, when you add the potentially MI contaminated bio-wheel. You've lost those three days and now actually begin again the copper treatment for 14 days from the time you put the bio-wheel in.
Another example: You've done your 14-day copper treatment and you are observing your fish for three weeks now (total 5 weeks in QT) to verify your treatment was successful. You want to put your now-prepared bio-wheel in the QT. No! The bio-wheel still needs to be operated another 3 weeks in a fallow system to make sure it is not Marine Ich infected. You don't want to risk re-contaminating the fish in QT.
I would shy away from doing the first example only because I would not want my fish to have to face a copper treatment any longer than they have to.
"Soaking" throws me!

To establish the beneficial bacteria in your bio-wheel, you need to actually
run it in the established aquarium for a few weeks. Is it running or just sitting there?
As long as the bio-wheel is running in the main display and the display system is fallow/fishless, then at the end of 8 weeks, there will be no more Marine Ich in your display and no more Marine Ich in your bio-wheel. In fact, there will be no more Marine Ich in your entire display system (sump, refugium, pumps, tubing, etc.). But you don't want to use the bio-wheel on your QT while it is in treatment, for at least 8 weeks, IMO.
After everything is Marine Ich-free, then the next marine fish or any specimen you quarantine, you can use the bio-wheel on the QT.
Did I answer everything? :rollface: