As soon as I saw your thread title I thought of converting an ice chest, insulated and has a built in drain. seems like you could mount it on the lower swim deck. The trick is keeping the water oxygenated and cool.
Your challenge will be keeping game fish alive as opposed to minnows, they need a lot of clean, cool, well oxygenated water to keep them alive.
When you were on
@adrianp89 fsh if you ran the aerator you may have heard a gurgling sound that sound comes from the MAX AIR set up. On the intake side of the aerator pump which is attached to the live well there is a small tap / hose hitting for a 1/4” piece of tubing that goes up to a small black holder mounted on the bulwarks where is sucked in mixed with the water and that’s where the bubbles come from. That’s one way to oxygenate the water, or you could mount a battery powered aerator on the side, I have a Frabill bait bucket that also can use a 12V plug in cord to run the aerator but that’s not nearly enough 02 for more than one wally.
Another way to keep the water aerated and fresh is to run the bilge pump as in
@viper15 design all the time, but instead of just moving the water around inside the cooler run a pick up line into the water from the internally mounted bilge pump and let the bilge pump pump water into the cooler and flow flow out of the cooler through a drain mounted at the top. The 222 / 252 / 255 FSH live wells work this way, no aerators like the 190 / 195 / 210 have. The advantage of doing this is that when you have a bloody fish it will flush the water continuously and keep it fresh and cool, I do the same with my 210’s live well. Set up your pick up line so it can be easily removed and capped so at the end of the day you can drain the cooler toss in some ice and take your fish home in the cooler.
Something like this.
But the challenge is the vane style bilge pumps need to be mounted at or below the water line, preferably below. They can lift water 6’ or so from the pump to the outlet or in this case live well, but they cannot suck water any distance to the pump. I asked my friend about your request and he told me a lot of boats used have a transom mounted live well pump below the water line with a quick disconnect.
I also remembered these bags from when I was a youngster.
SuperBags | Aqua World Bait Bags
You could use the ice chest design but you will need some way to keep the water fresh, I think your best solution is with a transom mounted vane style pump with a quick disconnect hose to be able to move the necessary amount of water.
Perhaps one of these tanks that could be easily mounted and removed
Edit:
So my friend and I were talking about your need for a livewell… we think you should put a plastic 20 gallon live well on rhe swim deck and tie it down, then install thru hull fitting on the bottom with a seacock. Run a 3/4” line to a vane style Pump for your live well fill / aeration, and tap off of that 3/4” line to feed an electric diaphragm style pump for a raw water wash down with quick disconnect in the bulwarks. You can also have a quick disconnect 3/4” line in the bullwarks for the live well source. Both of those pumps can be controlled by switches on the dash. The drain for the live well can just dump off of the swim deck. This way you can install your live well when you want to fish in 5 minutes, and you’ll have a raw water wash down on your boat always.
..... then the solutions would be two fold. Mount a pickup on the coolor with an external hose that I can secure to the transom when needed. Also a return hose right next to so that I can keep recurcilating the water. Also then put an aerator in it and mount that to the cooler as well. That way it is fully portable and can be used and removed. Put a couple of things on the outside of the cooler so it can easily be secured to the swim deck if put there. 
