Does a VSR current rating affect the size of leisure batteries that it can charge?
The Current rating of the PVR must exceed the Short circuit current of the PV in full sun by some margin. e.g. >  pref 30%. So if it is a 20Amp panel Ishort circuit then no problem.

the bigger problem

When the batteries are at different levels then the current connecting them together depends on battery resistance which depends on battery specs. e.g. if 1V difference and ESR is 5V/700CCA=7 milliohms then the switches must conduct 1V /7 mohms = 140 Amps.  (even if only for a short time)  

I used CCA test specs of a 5V drop at CCA level and I guestimated your CCA rating.

It is basically a MOSFET pair (for each battery) with a regulator and comparator to act as a smart switch with reverse protection for the PV.

When the engine is started and the start
battery reaches 13.7 volts, the VSR engages, allowing two battery
banks (start and house) to be charged simultaneously. When the
voltage drops below 12.8 volts (eg the engine is stopped), the VSR
disengages, separating the batteries.

So VSR rating depends mainly on battery capacity ESR and voltage difference when turning on PV above threshold.  There are tricks to get around this with a bypass relay and circuit but I wont get into that here.  ( Like a head lamp current limiter and bypass switch._)