"One or the other, not both BB and CV."
Then how do you explain combined carrier battleship operations off Guadalcanal and in the Eastern Solomons in late 1942? When battleships became available, about August 1942, they went with the carriers -- to Guadalcanal and to the Solomons, neither of which is particularly close to any fueling base. Yet the Navy operated both battleships and carriers in those operations.
None of the Pacific fleet battleships at Pearl Harbor were repaired in time to accompany the fleet before, IIRC, late 1942. The few battleships that were available after being hastily repaired at Puget Sound (Pennsylvania, Tennessee, Maryland) were held back and, during the battle of Midway, formed the core of the West Coast defense fleet.
They weren't held out of the Midway operation because the Navy couldn't fuel them. They were held back because they weren't fast enough to keep up with the carriers and cruisers and thus would have been a liability AND they were the only capital ships that were available to form a defensive barrier to a West Coast invasion in case Midway was just a feint. Remember, while Nimitz believed that the target of the Japanese operation was Midway alone, CNO Ernest King didn't agree and almost overruled Nimitz's plan to send carriers to defend Midway.
So yes, the Navy was more than capable of keeping both carriers and battleships at sea at the same time in the same operations and keeping them fueled.
Here's an interesting list of US Navy oilers during WW II.
https://www.ibiblio.org/hyperwar/USN/ships/ships-ao.htmlAt the start of the war, the Navy had roughly 30 oilers, the bulk of which were quickly shifted to the Pacific to support operations there. These, and the ships that came after, formed the backbone of the "Fleet Train" that made establishment of such forward bases like Ulithi possible.
https://www.cfla-alfc.org/stories/2017/2/23/the-us-navy-fleet-train-in-ww2Oh, and something else that I didn't know...
In November 1942 USS Maryland and USS Colorado formed the core of a battle group sent to the Fiji Islands area as a guard against Japanese incursions, and remained operational there for almost a year before they were rotated back.
For supposedly being kept on a short leash because of fuel concerns, that's a lot of operational time by those battleships, new and old, that were available in the Pacific.