Optimate 4 is £65 and you can use it for caravan/ car/ motorbikes etc.
Step up to 240v then back down to 12v is a very inefficient way to go!.
I'm sure our car charges the caravan battery through the hookup while towing, through the switched 12v supply pin..
Charges at 0.8A, 0.5A and 0.2, at £68 Lidi Ultimate Speed 3.8A, 3A, 0.8A, 0.1A at £14 the CTek is likely the most well know of the small chargers designed to recover lead acid batteries as will as the simple charging they do a range 0.8A, 3.8A, 5A and much larger models, the MSX 3.8 is 7 stage and costs around £66 clearly a lot better quality than the cheap Lidi.
However I am into day 3 charging with the cheap Lidi charger and so I would say for a caravan battery really too small, simple maths 3.8A the 0.8A pulse from 7.5 to 10.5 volt can't really be quantified and although it says it drops from 3.8A to 3A at 12.8 volt it does not give it as a percent of charge but graph shows around 40% and when it hits 80% it does drop to 0.8A so with a 88 AH battery if I assume it charges at 3.8A for first 80% it will take 40½ hours, at 3A it will take 45½ hours before going to maintain mode.
The OptiMate 4 has a 1A charge max and 48 hours run time max which is clearly why it stated 55 Ah battery max, after 48 hours what ever the state it will drop to 0.2A charge, so taking same battery 88 Ah after 48 hours there is 40 Ah left so it will take with no losses 248 hours to charge the battery. So in round terms a week and a half to recharge the battery. And this will no loses.
In real terms there are losses, and my cheap Lidi charger at 0.8A does not seem to be raising in voltage, not really surprised the battery is rather old, started Thursday just before 11 am at stage 2 the stage 1 was missed because I gave it a quick boost with a 12A battery charger to get voltage over 7.5 volt so it would see it as a 12 volt battery, it went into stage 3 without me noticing the step on the energy meter just after 3:30 Friday it went into stage 4. So 16.5 hours, at stage 4, the stage 5 is still running at 0.8A started with voltage dropping to 13.2 and now at 13.7 nearly 24 hours latter, as to if it will ever reach 14.4 volt needed to drop into last stage not sure.
As to charging off the car I have been there and done that, even with the second battery in the boot rather than in caravan the split charging relay would give it a boost after first starting maybe 10A but within 5 minutes it was down to 5A and for most of the journey around 2A, stick it in the caravan and in real terms it simply did not charge, OK may get 5A when car first charging but after 10 minutes down to 1A. Taking the three phase supply from alternator did work a bit better, I had two batteries charged that way in the works van with a 12/24 volt change over link and I used it to jump start wagons. But unlikely to find that in a private car.
So in real terms my 100 watt inverter and the Lidi battery charger plugged into that would recharge the battery quicker than the split charging relay. However as said 16.5 hours to get 80% charged, I do not tow a caravan for that long, not only caravans narrow boats have same problem, maybe cruse for 7 hours a day but even with twin 70A alternators one for engine battery and one for domestic battery it could not put back in what had been drained over night. With a caravan 30A is about max so in real terms all split charging does is extend how long before the battery is fully discharged it will not keep up.
The little generator throbbing away all day long is a pain, however it really is the only way one can run a caravan without mains hook up for an extended time, split charging and solar panels only extend how long before you need either a mains hook up or run generator. Main reason for not having mains hook up is in the summer it is expensive, in the winter so few use the sites no problem getting mains hook up for reasonable cost. So whole idea of charging from car is to save money, at £200 for a 30A DC/DC inverter plus wiring every time you change the car. We have decided for the moment not to bother, when we are no longer looking after my mum and we can go away for months at a time then it may change, but I think even then an inverter generator is a better option.
I do however thank everyone who answered it helped me work it all out in my head.