Is there any specific reason it would be too big of an advantage for a player if the benefits of Doubling Rings applied to a thrown weapon until that attack is resolved? Because I can't really think of any obvious ones. Both Doubling Rings and the Returning Rune are 3rd level items and the gold difference of 5 gp seems balanced by the risk of losing a couple mundane weapons that a character throws.
I suppose it provides a bit of an advantage to a character who carries a bunch of weapons with different damage types and traits and knows enough about each monster to grab the most advantageous one with their iron ringed hand and throw it. They would be getting a mechanical advantage over the returning rune, but they're sinking actions to draw them, maybe knowledge checks to grab the right thing, and money to be carrying around a coat full of hammers and daggers and shurikens and such.
I kind of like the idea of a character who just carries a runestone in the gold ring hand and uses the other hand to draw and throw weapons empowered by it.
Would this empower thaumaturges too much or step on their class features excessively?