The key here is that it skips on the small chainring and middle rear cog but not with the large chainring and the same cog. I find it unlikely and illogical that this could be a wear issue, as there is at least as much pressure when riding the large chainwheel and a given cog as with the small chainwheel, and many times people tend to push harder with the large chainwheel.

I think you need to recheck rear derailleur adjustment and alignment. Two things change when you change chainrings up front. First, the angle of the chain relative to both the front and rear. Second the angle of the jockey wheels change. The chain angle is relevant only for the front, as the chain is fed onto the rear by the jockey wheels. If the derailleur itself or jockey wheel assembly is twisted the chain may feed onto a given cog fine when at one angle but slightly off when at another.

You can check 1st to see if a small derailleur adjustment will fix the problem. Just do a partial turn either way and see if the problem goes away.

Even if that helps, get your bike up on a stand with the derailleur in the middle cog in the back and the small chainring in front. Look at the jockey wheels from the rear and see if the assembly is in a vertical plane (parallel to the wheel and stays parallel as you pull back on the derailleur. If it does not then something is out of line and you should take it to a shop experienced with checking derailleur adn hanger alignment. If they don't know what you are talking about go elsewhere.