Do you have a tap point on your 24 volt battery at a lower voltage - say 12 or even 6 volts - to which you may attach the low-dropout regulator?
If you are worried about unbalancing your battery bank, this may not be an issue: Some of the very low quiescent current regulators draw only a few 10's or 100's of uA - in the "noise" of the imbalance likely to occur between the cells of your battery system, anyway, and whatever controller you use, if sufficiently power-miserly need only draw a similar amount (or lower!) when asleep.
If you are still concerned by the operational power drain of the device unbalancing the system then one could actively switch using P-channel FETs, under processor control, to another, more conventional regulator operating from the 24 volt supply to get more current, perhaps a higher-voltage (15 volt) regulator diode-ORed with the low-dropout one: This would be done either during those short periods when one needed to wake up to perform some sort of management task, and/or the solar power budget was adequate for the task.
To go to an extreme - but it probably doesn't help with a PixAxe - if you have access to a point in the battery string where you can get just 3-4 volts after a diode drop you could run the PIC directly from that: A Nanowatt PIC would draw a negligible amount of power when asleep or ticking over at 31 kHz. Again, when full power was needed one could use some FET switches to connect to a higher-current regulator running from the 24 volt side.