I think I narrowed it down to my Inventory being replicated very slowly. As I said, I got the OnRep_Function to work and I update the HUD in there. This can take up to second, I guess I have some networking problem unrelated to the Inventory, since the same thing happens when I change my Health Variable. When I check the ping in PIE, it’s around 14ms so I guess it’s not that. Very weird.
**Update: **
Okay I think I got it, I’m such a dummy…
“MinNetUpdateFrequency” was set to “2”, which means when replication of a variable occurs unsteady, it’s throttling down the how often an actor is considered for replication (if I understood it correctly).
Two updates per second means I could experience a maximum delay of half a second, which is almost what I felt it was. 
Now I just set it to 20, which is probably overkill, but it updates very smoothly.
If someone’s reading this, I’ve got a followup question: How do you determine a good Value for UpdateFrequency variables? Does anyone have some resources on that? Does Unreal provide tools to measure bandwidth usage?
Information very appreciated but not needed at this point in development. Sorry for making such a big fuss of this problem, you can’t imagine the relief I’m experiencing right now.