Hi Jones,
I am so sorry to hear about your situation! It must be such a worry for you. I can't even imagine what you are going through.
I did have a baby who weighed less than 3 kg (sorry, I am not sure how that translates to pounds) though, and he was tiny but absolutely perfect and with no health problems at all. The ultrasounds obviously did estimate that he was small, and I gained much less weight with him than I did with my 3.7 kg daughter as well.
My doctor was not concerned at all, and everything but the size was just fine, and frankly even the size was just fine. Also keep in mind that ultrasounds can be off on size quite a lot.
In your place, I would absolutely be seeking a second opinion and also discussing the reasons your doctor is suggesting hospital bed rest. What will happen, in your doctor's opinion, if you don't go on bed rest? How will hospital bed rest benefit your baby? What does being on bed rest have to do with your baby's size? I am a bit lost here, and it sounds like you are not sure about the medical reasons for this suggestion either.
Get those answers, I say! Get answers before making these important medical decisions. The same goes for the next steps, which your doctor could suggest as a follow up, like having your labor induced at a certain gestational age, or having a planned cesarean section. If you are not sure how certain medical actions serve your baby, then get all the info before making a decision. You don't have to follow your doctor's advice blindly, not at all.
All the best,
Olivia