System to charge 24V lead-acid battery from a 12V lead-acid battery
It seems like it should be possible to set up a charging system that converts from 12V DC straight to 24V DC.


yes,  "possible".  You would do it with a DC-DC boost converter.  you can look up types of dc converters on the web.  At this level you basically take 12v @ 10 amps and turn it into 24v @ 5 amps, then you would just need some higher vdc than your battery to make current flow into the battery to effectively charge it (until you overcharge, cook, and ruin the battery).


  Ideally I'd like the 24V system to be some sort of intelligent charging system


with that you are likely going to look for a smart charger sold that will output 24vdc.  The typical smart chargers I think everyone is aware of are for 12vdc.  And for 12v it is not 12.0 vdc that the typical lead acid battery is, at 100% charge is about 12.65vdc and varies with temperature.  The battery at 12.20 volts is around 50% discharged, and at 12.0 volts is actually dead (not enough power left in it to be useful).  That would correlate with a 24vdc battery however the math goes, something like 2.1 volts per cell in the battery, and some differences between lead-acid vs AGM vs gel battery types.  The "smart" charger usually takes care of all that.


  converting the nominal 12V DC to 240V AC then back to (approximately) 24V DC to charge the scooter battery – seems grossly inefficient to me


yes from a strictly power usage perspective.  There will be however much inefficiency of the inverter from 12vdc to 120vac, and however from 120vdc to 24vdc smart charging.  But in terms of cost and availability of equipment to accomplish this based on having only solar panel output, it is not "grossly" inefficient.  And if you did the math for how much KWHr's are used in charging vs what is wasted from inefficiency it would be pennies on your power bill.

Any smart charger you would be buying is going to plug into 120vac.  I seriously doubt someone is making a smart charger that runs off 12vdc (solar panel output?) then smart charges a 24vdc battery.  If u asked this in the electrical engineering section someone would probably tell you what IC's to solder together to accomplish a simplistic regulated battery charging device.