Are the missed prayers due for re-fulfillment even with ignorance of the correct order?

You are not required to re-pray the missed prayers for two reasons: First, your method of making up prayers (Asr after Dhuhr, and so on) does not violate the order, but it does violate immediacy, which is obligatory according to the majority of scholars. The second reason is that the order between missed prayers is not obligatory according to the preferred opinion, which is the view of the Shafi'is and a narration from Imam Ahmad. Imam Nawawi chose that their ordering is not obligatory except with clear evidence. In summary, limiting yourself to making up one prayer with its equivalent violates immediacy, which is obligatory. Rather, you must make up as many prayers as you can each day.