A lot will depend entirely on the way in which William acts - he learned a lesson during the reform crisis when he refused to create enough new peers to get the bill through the Lords and Grey resigned and the King tried to restore Wellington and ended up having to reappoint Grey - his popularity sank and he was publicly humiliated - his popularity recovered (his actions were largely blamed on his wife and brother Cumberland a little unfairly to be true given William himself had serious issues with expanding the peerage too much in order to get the bill through).
But again his second attempt wasn't any more successful over the Russell affair as you mentioned - though in OTL he sacked Melbourne's ministry and Peel became PM in the November - but was forced to resign when he failed to gain a majority in the 35 election despite the blatant support of the crown - I am not sure there is a situation where royal support could ensure a Peel victory in 35 - which might set precedent that the King should sack government's at will in defence of the nation's interests- however let's face it his objections to the radical Russell weren't about defending the populace but about his own personal views of Russell and his politics.
Trouble was as mentioned the lack of senior Whigs in the Commons - given Melbourne was a bit stuck with Russell (and they were on diff wings of the party). Of course William eventually realised Melbourne wasn't that radical and they got on ok for the rest of his reign.
You might be able to ramp up extra difficulties between the King and Palmerston who really didn't get along at all lol.
Now active support from the regent Adelaide and the young King to Peel and the Tory's in the 37 election might help push them over the top meaning a Peel administration rather than Melbourne limping on - which might give the implication to the young King that exercising his influence can work and not attract criticism - how that works in the 40s with the growing Chartist movements, revolutions abroad, the corn law rows and the Irish famine is anyone's guess.