This might sound odd but a thought?
If he loves walking holding your hands, could you begin playing a game with him where you sit while he is standing but you are holding him under the arms then try a balancing game?? By this I mean make it fun to see if he can stand all by himself with no support and so when he does lose his balance for a little while you are there catching him but still allowing him to feel that 'free fall' sensation in a safe environment................does that make sense?? You catch him but allow a little fall sensation first (so just before his bottom touches the floor??)

I guess I am thinking if he finds this fun he might not be so worried about falling, if that is what is causing him to get upset??

Sorry, not much help but just a thought