[alistairsam]

hi,

if you have access to another battery, do try adding that in parallel first and checking if the same behavior is noted.
else, usually one car battery should provide ample power and their chargers are fairly inexpensive.

the capacitors won't give sufficient reserve as they only provide power momentarily depending on the load, so this could be from a few milliseconds to one or two seconds and largely depends on load draw. they're mainly designed to filter rectified dc and ripples in the waveform when used as filter caps.
but like I mentioned, they do increase voltage slightly even with load.
in Audio amps, usually 2000 to 10000mfd capacitors are used in parallel and they provide power during bass notes which require higher power to move the driver.

my suggestion is purely theoretical as I haven't tested this. I would try a load other than the laptop first that's rated at 15v.
measure voltage at battery before applying load, after applying load, and after adding 2 x 2000mfd caps in parallel, but the capacitors need to be rated at least for 50v. I don't remember the formula to calculate voltage rating.
one aspect to be cautious is discharging the capacitors. the higher you go, higher the charge retained. so a simple load like a 15v dc bulb with a 1w resistor and an spst momentary switch would help discharge them.

also try your original configuration with the inverter, but separating the batteries for the laptop/inverter and the motors. that way, if you slew at higher voltages, that won't affect the laptop, and keep observing which battery set drains quicker, and you'll get an idea of what AH rating you'll need and how many batteries in parallel for which.