Don't see and reason it would not work for short periods. The GW livewell has a small drain hole at the base of drain pipe that constantly allows the water and fish waste to drain and be replaced with new water. I would guess you could do the same if your Coleman has a drain at the base.
It may not be that simple but as mentioned for short intervals it might work out, First of all, the wash down pumps are generally rated for intermittent use not constant. It may overheat.
The livewell usually drains from the top of the water not the bottom. That maintains a set water level. By adjusting the rate of fill and the rate of drain, you may be able to maintain a fixed level but that won't be easy.
The feed is also usually near the top of the water to promote circulation. I guess that having the supply nozzle at the bottom would do the same thing
The most critical factor is the rate that the water is exchanged. That means how many times per hour the tank volume is changed. If two low, the bait will die from oxygen starvation. Too high and the fish will be stressed due to the fast current.
Of course if you use an adjustable nozzle, you should be able to adjust the flow rate.