Title. I have also explained this to the other mod author and made a ticket within their Github issues tab. They said in order for it to work, 1 simple function must be added from your end. The explanation can be found here -

https://github.com/TinyModularThings/Player-Weight/blob/master/playerWeight/api/WeightRegistry.java#L170

Is this at all possible?

Since backpacks don't store items in the item stack, I think it would actually have to be registerExtraPlayerInventory(), but otherwise this is doable.