Could Germany support its carriers with supply ships?

Would Germany build better supply ships to support the larger carrier and battleship force?

It should, but probably could not spare the yard capacity beyond the disguised supply vessels Germany already used.

German raiders historically relied on supply ships, including vessels operating under neutral cover, so the logistics concept is already present. A carrier force would benefit from more capable and purpose-built support ships. The obstacle is the same limited shipyard capacity already committed to capital ships and carriers. What the navy ought to build exceeds what its industrial program could actually deliver.