In most cases, if you have a leak in the fresh water plumbing, it will leak when using the pump like when using the city water connections. The main exception would be if the city water connection has much higher water pressure than the pump. So yes, you can put some water in the tank, turn on the pump, look for where the water is coming from and then switch off the pump. (I would recommend also turning on a faucet when you turn off the pump, to relieve the water pressure and make the leak stop faster).
I don't recommend trying to use the trailer this way (turning the pump on and off all the time, and having to mop up the leak every time you use some water).
To find the leak, I suggest drying it all out, and then putting paper towels all over the area. Flip the pump on for a moment, then off, and look for where the towel gets wet. You might be lucky and the problem could be one of the pipe connections you can see (which means you can get at it to fix it).
The worst case is probably the water heater tank having a crack. Are you sure you drained it when you winterized your bambi?
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