I'm going to chance my arm that there's no automated solution because there's no [realistic] automated way to find and trigger all the possible sound effects. You're probably going to have to be a human who plays for long enough to figure out how to trigger all the sound effects, in which case you almost might as well have captured them when you play.
Given that Atari 2600 games generally don't have music (i.e. Pitfall 2 aside), I can imagine a tool that automatically separated sound effects from a long recording, but that would probably be more straightforward by post-processing whatever you captured.
Top tip though: the audio counters inside the TIA are clocked at 1/38th the rate of the CPU.
So in the NTSC Atari 2600 the CPU is clocked at a third the speed of the colour subcarrier, so approximately 1193181.8Hz, and the TIA can update its output level autonomously at only 1/38th of that = around 31399.5Hz.
On the PAL (and, I think, SECAM) Atari 2600 that's 1182298/38 ~= 31113.11Hz.
I don't think that CPU changes to things like volume are synchronised to that clock and Stella's a generally excellent emulator so it's not quite true to say that sampling at 31400 Hz will exactly capture all NTSC sounds, but I think for this purpose it's close enough to true.