As above, why flac rather than mp3 for this application? Flac is lossless so your effectively size limited by the length of each recording you're encoding whereas with mp3 you could sacrifice bitrate to bring your file sizes in as required. You prefer flac but if you need to compress to get file sizes down then lossy compression becomes your only option - or split files.You could still convert the resultant mp3 to flac as before but why bother, you're not gaining anything?
I increase the volume to 999% (software limit)
doing math to calculate the highest bitrate possible that will still get me under or at 291 MB
The OP should probably look for a means of dynamic range compression rather than just increasing the gain, preferably range compression during recording.