Second on the giant beetle suggestion. In fact, unless you run into water elementals or kraken, you can probably take FIVE pack animals with you, but you can at least mount the giant beetle (unlike the others) if you have to go into combat. Shallow water fishing means you can just have a beetle with you as you travel, and if something aggro shows, you mount up.
All High Seas ships (Tokuno, Gargoyle, Orc, Britannian) have hold capacities in the high-4-digit or 10-12 thousand range (still just 125 items), AND you can fill and break commodity deeds in them (so if you hit the capacity in one of those holds from weight, you can deed the fish stacks up to free more weight up).
UOAssist with the classic client will allow you to stack fish at your feet if set up properly (put one of a fish on the ground, all other fish of that type will drop down and stack), but remember to move them to your hold or pack animals before they decay (as it takes about an hour for decay, a half-hour timer for packing them up works, and you can use things like a kitchen timer, online timer, or smart device to remind you). You can always pet-ball or gate your packies to the boat for said loading, based on the weight to be carried.
You only have to keep the fish whole If you are stockpiling the 12 shallow/18 deepwater fish (and later the 12 dungeon fish) for quests (to try for the power scrolls) or to sell to people doing quests. If you're just supplying a cook, you can cut everything up (4 steaks at 10 per stone, versus each fish at 10 stones).
Most "big" fish are not going to be the pretty wall mounts, so you can use pauses between macro iterations to cut up generic big fish for fish steaks, and the recipe fish for their fish steaks (as their specific fish steaks are the ingredient, not the uncut fish).