Somehow this actually sounds like a secondary problem to the gearset. The gearset appears to be functional, however outside forces are acting upon it.
Initially it is not working until the motor is warm...when cold not getting adaquate lubrication...then adaquate lubraication and works...then gets finicky again when hot. You either have a clutch problem (grooved basket, etc) or you are having a lubrication problem.
With 10 weight oil...you really shouldn't be having a cold problem, however heat may very well play into problems as these little motors get firecracker hot and 10 wt really isn't enough protection.
Is the oil multigrade oil? 10W30/40???
I have always run multigrade full synthetic oil in my XR's/little CRF's and have beat the holy snot out of them. Shifting problems have never been an issue unless there was 1) a clutch issue or 2) a lubrication issue or 3) a WAG here but shifter position actually makes a difference.
You didn't mention whether you have ever cleaned the two sets of screens out??? Partially blocked screens could account for oil flow blockage. Was there any debris in the screens?