Basically, the answer is yes and personally, I believe a synthetic is advantageous in this application. Whether it can reduce the temperature at the clutch plate in the engaging/disengaging process, is nothing I can promise. Generally, the transmission should run cooler and smoother.
Is this a semi automatic transmission of some kind?
If I remember correctly, the basestock does affect clutch plate friction, along with the friction modifiers of course. An optimal fluid would have well balanced frictional properties for the particular transmission. I would like to ask whether engine oil really is the very best choice here.
Can you find a ZF fluid spec. in your manual or on the identification plate?
Being a converter transmission, it might ask for an oil meeting ZF TE-ML-03, suffix A – H.
If you have a ZF model number, you can try to find it in the ZF oil list to see which suffix is recommended for your transmission.
ZF TE-ML 03 Converter transmissions for ...es, lift truck)
To me, it looks like UTF is preferred for most applications. Cat TO-4 type fluids can be found there too.
Does the UTF your are using in the shop happen to list a ZF spec.?
At the end, it is all up to you and if you want to use a synthetic engine oil, I would not see any reason to talk you away from it. ZF, on the other hand, does only recommend petroleum based engine oils in class 03A.