Minecraft 1.16.5
Forge 36.0.46
Lootr 1.16.4-0.0.4.16
Carryon 1.16.5-1.15.3.13

If you open a chest (turning the chest blueish) if you then pick up the chest using the mod Carryon it looks unopened. I have not tested this with barrels, but I am assuming it would do the same thing.

There's absolutely nothing I can do about the chest renderer in this instance, sorry.

And honestly, anything involving Carry On or chest movers is super-edge case. That I've fixed the dupe bug and added some mild support is about the extent of how far I'm willing to bend over backwards for it.