What is the ruling concerning the money that the questioner recently designated for his non-scriptural wife, and is he obligated to pay it given her retraction of forgiveness?

You must sincerely repent to Allah for the sin of marrying a non-scriptural woman, an issue on which there is no disagreement among Muslims.

As for the deferred dower (sadaq), if the wife willingly absolved you of it, then her right has fallen, and she cannot reclaim it, because what is relinquished is as if it never existed. The reason you mentioned for her waiving her right is irrelevant as long as she willingly waived her right.