If that was the idea, they’d probably pick something that we haven’t publicly copied.

Basically, yep. This drone is designed primairly as a target drone; since one of the things the US most wants to be shooting down is Shaheds, the form and capabilites are very similar for training use. (It’s supposedly a more advanced version, anticipating developments that Iran et. al. will likely (or have already) rolled out. Who knows how true that is, though.)

The Russians have already integrated jet engines and some other enhancements into their Shahed clones, Geran.

Are the Russians upstreaming their improvements back to Iran?

That’s a good question, I am not sure