I assume, when you say pairing ring you meant to say doubling rings?
If that is the case, I see no reason why not. As long as you wear both rings and hold both weapons, the fundamental runes on the weapon held in the hand wearing the gold ring are temporarily copied over onto the weapon in the hand wearing the iron ring (suppressing the second weapon's own fundamenal runes - if any - in the process).
At no point in the process does material play a role.
PF2 doesn't impose any limitations on Runes that I can find based on Special Materials like PF1 did. The only material that has anything to do with runes is Orichalcum which can bear 1 more Property Rune and halves the cost of a Speed Rune placed on it.
So as albadeon said, Materials don't matter when using Doubling Rings.
Purity of special materials places limits on runes.
Low-grade items can be used in the creation of magic items of up to 8th level, and they can hold runes of up to 8th level. Standard-grade items can be used to create magic items of up to 15th level and can hold runes of up to 15th level. High-grade items use the purest form of the precious material, and can be used to Craft magic items of any level holding any runes. Using purer forms of common materials is so relatively inexpensive that the Price is included in any magic item.
The question is whether the doubling rings can impose a high-level rune on an item which, due to its impure-grade materials, is unable to hold such a rune.
Purity of special materials places limits on runes.
Low-grade items can be used in the creation of magic items of up to 8th level, and they can hold runes of up to 8th level. Standard-grade items can be used to create magic items of up to 15th level and can hold runes of up to 15th level. High-grade items use the purest form of the precious material, and can be used to Craft magic items of any level holding any runes. Using purer forms of common materials is so relatively inexpensive that the Price is included in any magic item.The question is whether the doubling rings can impose a high-level rune on an item which, due to its impure-grade materials, is unable to hold such a rune.
Neat. I was not aware of that. In that case, and after reexamining the situation I would tentatively agree with albadeon still. Doubling Rings does not require a weapon to meet the requirements of the Fundamental Runes to be applied. Greater doubling rings require that the iron ring weapon, "meets all the prerequisites for a given rune and is not a specific weapon." but that is referring to the Property runes being replicated by the Greater Doubling Rings.
The description of the Rings does mention "replicating" the fundamental runes, which to me sounds like they sort of apply some form of temporary etched runes on the receiving weapon. However they do not require them to meet any prerequisite besides being wielded in the Iron Ringed hand.
So I suppose my answer would be a firm, "Yes?" and shoulder shrug.