With the engine running the alternator feeds the fridge; when it stops the load falls on the battery. A fridge whose label says 150 kWh a year draws about 0.4 kWh a day, while a 12 V 100 Ah battery holds 1.2 kWh in total and cannot be used down to zero. A starter battery is for cranking the engine with a high current lasting about a second and cannot be deep-cycled, so this job needs a separate service battery.
The fridge's climate class is a limit too. According to the Bosch refrigerator manual, class SN runs at +10 to +32 °C and class T at +16 to +43 °C, and the hot air at the rear panel must not be blocked; compare the body's summer temperature with that range.
During transport the compressor oil may have flowed into the cooling circuit; the same manual says to wait at least 1 hour after installing before switching the fridge on.
The option that does not depend on electricity is an insulated container with eutectic plates: in the GEBHARDT Isotec TBX the plates are frozen beforehand in the depot, the container draws nothing from a battery on the road, and with eutectic plates GEBHARDT keeps the cold for up to 24 hours. To decide, write down the longest wait on the route, its hottest hour and the total time the engine stays off; which solution is enough follows from that.