I have a Cavern Club SE in blue oyster pearl-20, 12, 14, and it is a wonderful sounding kit, very round for sure. My snare was a bit fussy in getting the snares to resonate properly, but I've got it tweaked now with Pure Sound 30 snare wires. It sounds very full and woodsy with either an Ambassador or Emperor batter, and a Hazy A resonant.
For projection and to 'keep the tone alive,' I've found a stronger sound on the toms using an Ambassador/Vintage Ambassador for the batter.
My current combo is an A batter/Hazy A reso for the mounted tom; and Vintage A batter/Evans Strata 1000 resonant on the floor tom. The thicker Strata gives an even fuller sound than a hazy A did. I chose those particular resonant heads based on what I had on hand, but they work well.
I've got all the drums tuned to medium high resonance using a Tune Bot.
They sound large and round and cut through just fine.