This comment from A7pr8 seems to suggest it is possible to modify the apk via Lucky Patcher, which hopefully would then work on a non-rooted phone:
You can't just send the app from your rooted device. When you mod the app on a rooted device only .odex file will be patched not the .apk file.
You need to rebuild .apk to be able to share it and the the other device should also has luckypatcher for it to work.
Simply when you patch an app use the option create modified .apk or rebuild it from "rebuild & install" in the bottom bar.
As someone who just switched to an Xperia as it's one of the few phones that still has a headphone jack, NFC, and microSD slot, this sucks. The software support is the biggest problem IMO because the rest is great.