No, the Steam versions and the non-Steam Windows versions aren't compatible. Technically they're exactly the same (same download client, same updates from Square Enix), it's just a licensing thing.

Your only option is to buy the expansion from Steam, although don't lose hope as they recently had a sale on HW so they're likely to again in the future too.

Do not buy a new copy of ARR reborn though, that will be for a new service account/new subscription and new characters. You can only apply one copy of each platform (Windows, PS3, Mac, etc) to each service account and since Steam and non-Steam PC are both classified as Windows, you can only have one of them.

Hope this helps, even though appreciate it's not what you wanted to hear.