Disney Infinity 3.0 cross-platform playsets and characters missing

I downloaded Disney Infinity 3.0 game on Windows and bought the Rise Against the Empire and The Force Awakens digitally, and got my kids a couple physical characters. For Xmas I bought my kids an Xbox One so they'd let me have my computer to myself, and got the Twilight of the Republic starter pack, assuming it would complete our set. But after installing the game on the Xbox, it will only let me play the playsets for which we have a playset piece that we can stick on the base, and only the characters with physical toys that we can put there. I've searched all over the web to investigate this issue and I can't find a single mention of not being able to use "web codes" on the Xbox. It doesn't make any sense to me. Why are there "web stores" and online accounts if you can't use them on all platforms? Are there any options for me other than double-purchasing the RATE and TFA playsets that I already spent $50 on? (What else is a bummer is that we spent hours trying to kill the AT-ATs on Hoth before we actually cleared the zone...)