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You can in theory. However, it is likely that your mobility scooter's battery will be larger than your leisure battery, so it will soon go flat. You'd need a tonne of solar (and only charge in sunshine) or EHU really. If being able to do this will be transformative for you, it may be possible to re-design your electrical system to enable it. However, it will be expensive to do so, involving large lithium battery banks and lots of solar panels/large B2B chargers and MPPT controllers.
I agree with cmcardle75, a big lithium battery connected direct to the alternator with a split charge relay might well cause problems. This can be solved by using a Battery-to-Battery (B2B) charger which will control the charging from the alternator to the battery. It will avoid overheating the alternator, and will properly charge the lithium battery.
A 60A B2B charger will output 12 x 60 = 720 watts. A big battery, perhaps 1440 watt-hours capacity, will take 2 hours of driving to recharge from flat. If the battery is smaller, and/or you are happy with longer hours of driving, you could use a 30A B2B. For comparison, a 1440Wh battery is equivalent to 1440/12 = 120Ah, about the size of an average leisure battery.
Edit: just seen your post. A 37Ah battery will charge OK with a 30A B2B in just over an hour of driving, if it's 12 volts. Is it a 12-volt battery? That is between 12 and 14V?
Sorry, should have read question. If you already have an inverter fitted in the motorhome, then you can use it to charge the mobility scooter battery, with the scooter battery mains charger. You'll have to check that the inverter is powerful enough (how many watts is it?) but I'd imagine it will be OK. Do you have the info from the label on the scooter battery mains charger?can I charge a lithium battery via my inverter?
Is your inverter a pure sine wave type? Any other could be detrimental to your mobility charger electronics.That’s good news & very reassuring. CareCo recommend charging after every use. As we usually mover every couple of days and drive for around 2/3 hours so should be ideal. Thank you to everyone for your advise. Really appreciated.
I was not being critical, sorry if it came across that way, just trying to determine if the 240VAC battery charger you intend to use for you mobility battery will be ok with your modified sine wave invertor.As a complete dunce where electrics are concerned i thought that this would be the best place to get impartial advise.