If Synchromesh is recomended, Synchromesh is what you should use. It's similar to 5w30 in viscosity, but with different additives. 10w40 doesnt have the additive package so dont use it.
Go with GM's advice, not "some shop".
If the oil you guys have access to is GL rated... then GL5 is for diffs etc, GL4 is for gearboxes.
GL5 has anti-wear additives that will attack yellow metal synchromesh parts. It can only ever be used in manual box if the box has steel synchro rings.