Odd that the pedal works with the engine off. This makes me suspect the brake booster.

If it were my car, I'd start by bleeding the brakes and checking for visible leaks through the whole car, but especially around the master cylinder/booster and also around the pedal itself.

Next, I'd double check the fluid level and mark it before starting the car (I use a grease pencil on the outside of a clear master cylinder reservoir whenever I do a brake job and fluid flush, a little line can tell volumes down the road...)

Now start the car and pump the brakes. Does the pedal stay "soft"? Do the brakes work well or do they fade out? Has the fluid level dropped after a few hard applications with car running?

If there's air in the system, the pedal should be bad, engine on or off. If the master cylinder is bad, the pedal should slowly drop, engine on or off. If the booster is bad, the pedal should work as though there were no or weak brake boost, but fluid may be getting pulled past the seals, causing the pedal to drop--this can show as a loss of fluid in the reservoir with repeated pressing and holding the brakes with the engine running.

Good luck.