I began by trying to do the usual noise reduction, but I couldn’t find any sample of pure noise to use for the profile. How can I reduce noise in that situation?
Noise reduction works best on constant, low-level, background noise. If the noise is bad, the cure can be worse than the disease.
There’s probably not much you can do, although equalization may help “improve” the recording to some extent. (You’d just have to experiment with EQ to see what you can do.)
As you may know, on-location movie dialog is re-recoded in a (soundproof) studio because even with the best pro-software, there is only so much that can be done.
…Live recording is NOT easy. There are various techniques depending on the type of music and the venue etc. Big rock-shows are close-mic’d and multi-tracked. i.e. With a mic right in-front of a guitar amp, and close-mics on all of the drums & cymbals, etc., you can get a very-strong “signal” for a decent signal-to-noise ratio.
Acoustic music in good room can sometimes be recorded with a pair of microphones, but the mics have to be closer than the ideal audience listening position for more direct-sound and less room-sound.