Is there a clever solution to Arnold's "merchant problem"?

At the end the tea cup is as full as at the start. This implies that the added wine is exactly outweighed by the tea that has disappeared.

The volume of spoon, $s$, is the conserved quantity. It is also the amount of wine in the cup.
When you then take some mixture $\mathit{tea}+\mathit{wine} = s$ into the spoon,
$s-\mathit{wine}$ is the amount of wine left in the cup and the amount of tea poured into the wine barrel.

To a first approximation, there is a spoonful of wine in the cup and a spoonful of tea in the barrel. How much are each of these approximations off by? Well, there is a bit less than a spoonful of wine in the cup, since a bit of the wine was removed in the second step. And, there is a bit less than a spoonful of tea in the barrel, since there was a little wine mixed into the spoonful that was put into it. But these errors are exactly the same: both are the amount of wine that was in the second spoonful. So the two quantities are the same: both are one spoonful minus the amount of wine that was in the second spoonful.

Or, here's an even slicker way. Notice that the total volumes of liquid of the cup and barrel have not changed, since the two spoonfuls they exchanged cancelled out. So, the overall change must be that the barrel exchanged some volume of wine for the same volume of tea from the cup.

Note that your solution is actually wrong--when you compute the amounts of wine and tea in the second spoonful, you are assuming the cup was mixed uniformly after the first spoonful, which the problem tells you not to assume (that's what the "(nonuniform!)" is all about).