Well, for a short while-(during the 400's & 500's), the Northern Italian city of Ravenna, was, in a way, a type of quasi Byzantine capital...that is to say, Ravenna, during the early Middle Ages, was a type of, Byzantium of the West.
The Church of San Vitale, for example, is one of the best preserved early Byzantine style Cathedrals. It was built around the same time as the Saint Sophia Cathedral in Constantinople and was commissioned by Emperor Justinian.
There is also a surviving Byzantine Palace in the city of Ravenna dating back to the reign of Theodoric the Ostrogoth-(around 1600 years ago).
As to why Ravenna was the Western capital-(or Western counterpart) to Constantinople remains unknown.
With regard to relocating the Byzantine Empire to city of Rome....this would have made no geopolitical or commercial sense.
Constantinople, was the ideal location for the Eastern half of the Roman empire to thrive and succeed; it was (and is still), a bridgeway between Europe and Asia, plus it had-(and still has) access to major waterways, such as The Black Sea, The Bosporus and the Dardanelles/Hellespont, which lead directly to the Aegean and Mediterranean seas. In terms of commerce and geopolitical positioning, the city of Constantinople had advantages which Rome, was not able to match.
While the city of Rome had access to major European land and river routes, as well as the greater Mediterranean sea (and even the Straits of Gibraltar), its proximity to Asia-(and its famous Silk Route), was more distant than Constantinople. The Silk Route-(from West to East), began on the outskirts of Constantinople and ended in Xi'an, China-(Central China). Having direct access to the world's longest and most prosperous trade route, would have been very important to the success and longevity of your empire. The (Western) Roman empire lasted about 500 years.....whereas the Byzantine Empire lasted for over 1000 years.