I am aware that ATF is gaining ground for gearbox lubrication - my Jeep Compass uses ATF4 in a 6-speed manual gearbox, and we notice an easy change without crunching, although with 20,000 miles only recorded it had better still be in as-new condition. However you try putting ATF in certain gearboxes, my old Volvo F10 pulling 45 tons loaded with concrete for example, with a 3-speed synchro box plus range change and splitter, and I doubt it would run long before total gearbox failure occurred within a matter of hours. It has an oil temperature control system using the radiator refrigerant and uses SAE90/140EP.
The makers recommendation is always the one to go for, unless circumstances push you in some other direction such as excessive whine or poor synchro operation, and what you choose to replace it with is a matter largely of personal opinion. However these old Fiat boxes are hard to get spares for, so making do with what you have and taking care is the usual case.
I chose SAE90/140EP for my slightly worn box because it's what I read recommended in some tech book or other, which warned about using it where phosphor bronze components may be met with. I don't think the Fiat box has any, the spare box I have and took apart hasn't any anyway. Our climate is as hot and dry as yours reaching 110+ at times in summer, in Las Negras, Almeria Province, Andalucia, Spain. But a motor-home may travel to Arctic climates too, not to be forgotten.
Because the Fiat gear linkage is such a mess, slow and steady changes are called for anyway, which should help the worn synchro, if indeed it is worn so much.