The whole point of having certain things in limited supply is to maintain scarecity. The limited fleet bank tabs force you to choose carefully what you store in there rather than collect up all the consumables in the game and have an endless supply tucked away for your fleet's use.
Use what you have more carefully - think of it as a challenge
Start opening up tabs in your fleet bank for people to pull stuff out of. If you're one of those fleets that just stockpiles tons of stuff without ever giving anyone access, then what's the point?
I have ranks in my fleet for every rank in the game, and then our officer rank, and the fleet leader rank. I set up a tab for each rank, and set only people of that rank to be able to pull from that slot. 4 stacks a day.
Then, I have a tab for data samples, 6 stacks a day, everyone has access(because we have so many). I also have a tab for "rares", which is locked out to all but the officers and fleet leaders, and a consumable area for food, shield charges, and tribbles, that has full access to everyone.
By ensuring people are promoted properly, my fleet bank never fills up. And, really, it's not that hard to come by a Mk IV Dual Phaser Cannon [CritD], so when someone hits a rank and decides to get their ship upgraded, I don't feel cheated. We've got the rare tab for people who want to drop stuff that they don't want grabbed up by the first person to see it.
But, yeah, make sure you're giving your general populace ample opportunity to benefit from the bank.