The inventory system is a no-win scenario. It's responsible for the largest data blobs sent over the network, that's why Yoshi-P doesn't want to add any more. The recent change to the search was one way to reduce the amount of network traffic by not doing any network traffic and only searching what is already cached by the client. If it was on-demand loaded like some older games, every time you open, close, or drag something in, or use a consumable in the inventory would result in refreshing the complete inventory. Hence taking the smaller inventory inside the dungeons reduces the loading time to just those pages and the armory.
Since there is is the potential to fish inside an instance, the fishing gear has to be something in the players main inventory.
Now there is nothing to prevent SE creating an "ammo" slot where fishing bait, machinist bullets and archer/bard arrows could be stored. Or they could just remove how bait works to begin with and build it into the level of the fishing rod. Higher level rod/gig, higher level bait becomes available. Or maybe integrate that inventory on the fishing log, so that any crafted bait can be selected from there, or when you start fishing, so you know which bait works where.
Of those, obviously the easiest thing is to just remove bait, period. However I feel competition for the off-hand slot between DoL/DoH jobs is an issue and perhaps SE should see about separating DoL/DoH armory items from DoW/DoM, since the only place outside the overworld currently that DoL can be used is Diadem. There's no need to take DoH/DoL gear inside a combat duty, and that would reduce the amount of data that could potentially be lost.
If SE wanted to make a DoL/DoH-specific inventory akin to the chocobo storage, that wouldn't be unreasonable either, but it's still data that has to be synced frequently on the server side. So I imagine they'd be small inventory spaces that can only store items usable by that job (eg bait could only be stored in the fishing inventory, cooking ingredients in the cooking inventory, the completed cooked items would require main inventory space, or could be moved to the chocobo if not needed in duties.)