We have done this on a handful of occasions, most recently twice in Barbados a few months ago! We had to go 'out of the port' through immigration, and for our first ship transfer from Azamara to Windstar we sat in the covered outside waiting area with luggage for a couple of hours before checking in.
Returning from that cruise, we boarded Emerald Azzurra same day. Because that ship was closest to the cruise terminal, literally a 30 second walk, after 'exiting' through immigration I wandered over to the ship. Encountering the most pleasant Hotel Director on the dock he agreed to take on board our luggage. That left us free, and totally unhindered with luggage, to explore Bridgetown for a couple of hours before embarkation.
What you suggest will work perfectly in Civitavecchia, and you may find that Cunard (especially for QG class) will take your luggage immediately on your Seabourn drop off at the Cunard dock. Best wishes, I am sure your plan is sound.