As I tried to point out in my original post, it sounds like a broken return line - that's where I'd look first.
The pool looses a heckuvalot more water while the pump is running, therefore water is being FORCED through the leak.
If the filter system is below water level, it could be backwash blow-by - I don't think so though (but I can't put my finger on why not

) Also a pump problem would have been noticed.
A suction side leak would draw air into the pump, causing priming problems and bubbles, certainly not exacerbating the leak.
A stuck hydrostat valve or a cracked skimmer housing would cause water loss but, the amount of water lost daily would remain the same, whether or not the system was running.
It's always possible that you have multiple leaks in the pool, but given the discrepency between the water loss with the pump running vs. without, I'll lay odds on finding the main leak on the back (effluent) side of the pump
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