I think I know what's causing the bass loss as you up the gain. The capacitor in series with the gain pot makes it a low-pass filter on the feedback. The amount of signal going to the - input of the op amp is determined by the ratio of the 1 M resistor to the impedance of that shunt to ground with the cap and the gain pot. When the gain pot is at a lower resistance, less of the signal goes to the - input (ie higher gain) and the impedance to ground is determined primarily by the capacitor, sending the highs to go to ground and the lows to the - input. Increasing the resistance on the pot both sends more signal to the - input and makes the capacitor less significant in the circuit so the distribution of the highs and lows going to the - input is closer to that in the output.
You may want to play around with different values for the cap (larger ones will increase the bass response as well as increase the overall gain - a large enough value should pass everything in the audio range without any noticeable filtering), putting a pot in the position of the 1 M resistor to control the gain without altering the filter, and putting a small cap in parallel with the 1 M to send more highs to the - input like you mentioned. Putting a pot in series with that cap will give you an adjustable high-pass filter on the feedback.