Plumbing – the correct way to fix a leaky drain trap that is not detachable

Go with your second approach of getting the right size fitting except don't bend the pipe, just deal with the angle. You pan lip should be higher than the ending (with fittings) top of the drain pipe so it can drain but that's all you need.

Why? PVC fumes are extremely toxic and you have a significantly better chance breaking the pipe than fixing the angle; then you'll have to tear up the floor to fix the pipe. A washer machine drain pan is used for worst case scenarios, not day-to-day use so as long as water can drain you'll be fine and a slight angle may actually decrease the minimum height of the drain anyways.

As for drilling the hole in the pan, mark it in place and drill it somewhere else with a sacrifice piece of wood/etc. underneath. You don't want to hold it up or anything because you'll need a decent amount of pressure to hold a bit in place and plastics like to fracture if not properly supported.

I think the fact that a "dark damp patch appears on the concrete" when you open the shower valve(s) yet catch the water in a bucket points directly to a leak from the plumbing. You need to open the wall from behind to gain a view of the shower valve body, and also remove all trim on the inside. Use the handles to open the valves and find the leak source. Water may be running down the interior of the enclosure wall. The fact that you have a plumbing leak does not mean that your shower pan is in good shape; if it is old, since you have removed the tile, this is an appropriate time to break out the concrete and have the pan refurbished (hot-mop, vinyl liner, fiberglass pan, etc...)