Fleet supply ships are typically either armed merchantmen or else custom designed ships. I would say the Imperial navy would use the latter. A single ship would be designed to support a destroyer flotilla, or cruiser squadron and supply parts, ammunition and make repairs short of requiring a yard.
One should be at least 5000 tons, if not 10,000, depending on how much fuel is required for its range. It doesn't need to be fast, and it's not going to have much more than self defense armament. It should typically take up duty station for 3 to 6 months and then return to port for resupply. Crew will be split between the normal ships credit and all the engineers and support personnel who do the work. You coukd even have a dedicated command section for higher ranked commanders if there is no port available.
Small craft would include engineering support vessels, plus smaller ships to distribute supplies to ships not directly docked to it.
The ship would have a fabrication section, electronis repair shop, enginger ingredients repair ship, small craft repair shop, spare parts cargo hold (s), ammunition storage, food and other general supplies hold (s).
More than likely it will support and repair ships docked both port and atarboard, possible even dorsal and ventrally, but that may be getting too busy. Each side would have airlocks, cranes and repair gantries to facilitate repairs and make it easier for personnel to go to and from.
There is a wealth of information out there for how navies have done this over the last century.
