Don't forget that a normal AGM deep cycle battery should not be regularly discharged below 50%, whereas a Lithium can be regularly discharged to ~20%
Let's say (just guesses - you should measure them):
Your fridge takes an average of 2.5A.
Your cooler takes 1A (in the conditions that you plan to use them - I reckon a fair bit more for both in WA),
and you have them on all day (min 3.5A). That's a total consumption of 84Ah over 1 day (without detectors).
Now throw in charging say 2 * 10Ah (estimated) detector batteries each day (each say 60% discharged), that's a total of 96Ah each day.
To replenish this (without your detectors), you would need a 200W solar panel (with MPPT charger in the WA goldfields) giving say 10A over 9 hours (useable daylight - and only if moving the panels every few hours) or 90Ah charge during the day. Clearly this is not enough (with detectors thrown in), and you would need to run the car for ~1/2 hour each day (to be safe) or get bigger solar panels or a more efficient fridge/cooler.
Now let's consider the nightly discharge.
Let's say the fridge and cooler take 80% of the average current at night or 0.8*3.3=2.8A and the solar down-time is 24hr-9hr = 15hrs, that's a drain of 2.8*15 = 42Ah.
A 100Ah deep cycle would handle that well on good solar days. But now include the two 10Ah detector batteries (say you ran them 60% each day) and you now have 54Ah nightly drain, requiring a minimum of a 110AH deep-cycle Lead Acid AGM battery. Solution: run the car for the 1/2 hr a few hours after sunset - just to be sure.
The above assumes lots of things, like worst case current draw (on a hot day), worst case charging (cloudy day or significant shading), useable sunlight hours (Winter or Summer). You may need to go without the cooler or run your car for longer each evening or open your fridge less. Only careful measurements or experience in the field will tell. Check battery remaining charge each evening, and run the car's alternator accordingly. This is not being anal - it will save you cash and down-time if done right!
PS: if you have the cash, go for Lithium Batteries (~80% more discharge power and ~50% the weight of AGM Lead Acid batteries of the same size plus a lot more life cycles, but around 4* the cost) and a 250W solar system. Then forget ever having to run the car's alternator except on the hottest & cloudiest day.
No guarantees by me for correctness here.