The current setup – converting the nominal 12V DC to 240V AC then back to (approximately) 24V DC to charge the scooter battery – seems grossly inefficient to me
It is. It's intolerable. The inverter has "standby" draw simply for being spun up, which will bottom out your battery, which is damaging to lead-acids.
Cannot be done sensibly.
It seems like it should be possible to set up a charging system that converts from 12V DC straight to 24V DC. But I'm utterly failing to find a sensible way to achieve that.
Because it cannot be done sensibly. Battery charging is an extremely complex science. Proper charge management is essential or lead-acid batteries will be destroyed. Nickel-cadmium is more resilient, but still, you can't just slam a charge onto it higgledy-piggledy. Lithium batteries are the worst of all; mismanagement of their charging will lead to explosion and fire.
Battery charging must be left to the professionals, or to be more precise, to professionally designed chargers rated for the equipment. If this equipment has an external battery charger, with the smarts in the external charger, it's simply a matter of going back to the manufacturer and saying "Yo, I need a car charger for this thing, here's my VISA card". And they say "okeydokey" and ship it off to you.
Or it may be possible to find competent COTS 3-stage battery chargers able to input 12V-ish. Your 12V system is, after all, quite similar to a car's electrical system. Charging a mobility scooter from a car is a common enough application.
Then you wire a "cigarette lighter" style socket raw/hot into your battery, and Bob's your uncle.
Or use DUMP.
If the scooter does not need to be charged on-demand, and your solar charge controller has a set of DUMP terminals, it can be feed off DUMP. DUMP is energized when the controller has excess power it can't use (because the battery is full and charge energy exceeds all available loads). So the scooter charges only after the battery is fully charged, using only solar power that would otherwise be thrown away.
For that matter, the inverter could be placed on DUMP, and it would only create inverted AC with excess power.