IF the immersion heater in the calorifier is (we assume) 240v and is run from either your generator/shoreline, then the length of time it will take to heat the water depends on how many Kw it is. A 1Kw unit would take about 3 hours to heat the calorifier, (and a 2Kw unit about 1 1/2 hrs).
You definitely need to determine whether the engine will heat the calorifier (though I suspect that as an indirect raw water cooling system it may well not) .
New batteries and bigger inverter are certainly easy to fit, as are solar panels (though you may well find that a new solar controller is necessary depending upon the type of controller currently fitted.
In truth, almost everything is 'do-able', but always at a price of course.
To address your question of will the calorifier keep water warm during the day - the answer has to be - it all depends upon the insulation round it. Your description of a blanket wrapped round it doesn't exactly inspire confidence in the insulation, although it would be very easy to box it in, with generous air-space around it, (even a simple thin ply boxing would suffice) - and fill the space with rockwool, or polystyrene beads. (Or wrap it round with bubble-wrap.)