It would be great to see some measurements just to see what you are dealing with in the way of room modes. Can you share an .mdat of left and right channels measured at the LP? Don't gate the measurement, just use REW's default window of 500ms as we do want to see what is happening down low. Yes, I get it is only in one position in the room, but the reality is the room modes are (pre)determined by the physical dimensions of the room and by moving the mic all you are doing is measuring some portion of the peaks and valleys of the room modes, but they are still there
Wrt rePhase, while it is possible to use rePhase to correct for room issues, it is intended to be a loudspeaker design tool. OTOH, proper room correction software is designed not to boost nulls, but reduce peaks and fill in some of the nulls based on the transient response analysis of the room. The result is a more even bass response over a wider sweet spot and not over correcting at a single position. Of course you can take measurements at multiple positions and then average, but I have found the transient response analysis algorithms used in some room correction software does a great job of averaging with a single pair of measurements with multiple measurements at different positions offering no additional benefit. OTOH hand some room correction software like Dirac Live, do require multiple measurements as it is using a different analysis algorithm.
While neither here or there, the place to start is to look at the measured response as currently setup and then based on some analysis, see what the possibilities are.