How does upgrading my supporter pack work?

I bought the early access through steam, but am considering upgrading.
If I buy the upgrade through the in-game store, will I also get it also get the DLC on steam or should I just wait for the prices on steam to be adjusted?
I like for it to show everyone that I own the pack.

You can upgrade through the store on this website. From what I see there's no upgrade functionality for the packs on Steam