How to fix a minor leak in situ
Hire a water tank truck. Some pool supply stores have them; or depending on the state of your community you might be able to strike a deal with the fire department, subject to no fires happening at the time. Park the (empty) truck, pump the water into it, fix the outlet, pump the water back into the pool. The fire department may be able to treat it as a training exercise; in other cases they are simply happy to help, and in yet others they will not help; it varies with the community. In several places I have lived they are the default pool fill service, and it also means they know where to grab a few thousand gallons of water in a hurry  when there are no fire hydrants in the area...

Depending on how good you are at breath-hold diving or whether you know someone with scuba gear, you might also try applying your appropriate type of grease around the outlet from inside the pool. Taping outside seems unlikely to work, IME.

Thinking it through a bit further, (if it's a bottom drain) find/buy a small innertube. Take a 5-7 gallon (20-30l) bucket and try to find a tube just a bit smaller than the bottom of the bucket (assuming the bucket is larger than the drain - otherwise you'll need to upscale the bucket.) To reduce hilarity, remove the valve-stem and fill the tube with water, then replace the valve-stem. Put rocks or bricks in the bucket. Place the innertube around the drain and the bucket on top of it. Drain some water and see if it seals.