What if too many Movements/Fatigue (exceeding the Resistance of the mule) caused the monk to fall off his noble steed shamefully ?
He could get start with some "Pride" tokens and lose them one after another during the race...
I like the push your luck element inherent with the yahtzee type dice, but also the fact that it is directly affected by the mules overall stamina - perhaps, instead of the mule skipping a movement turn (as suggested in the original post)or the monk losing pride points, you could try having the mule only refresh up to half its stamina the following turn.
This would give an opportunity for some clever card play with the types of cards that add stamina for the round, or perhaps cards which actually confer an advantage if you have not used up the mules full movement for a turn (thematically, think the monk stopping for a roaring festival in a village and even gaining prestige/pride points by doing so - another thing this brings up - could there be alternate paths to victory here? Prestige vs. Race winning etc?)