How can the Hadith "Were it not that I would overburden my Ummah, I would have ordered them to perform ablution for every prayer" be reconciled with the Hadith "No one maintains his ablution except a believer"?

There is no contradiction between the two hadiths. The first hadith indicates the desirability of renewing ablution for every prayer for one who is not in a state of minor ritual impurity, while it is obligatory for one who is impure to perform ablution. As for the hadith: "None maintains his ablution except a believer," it encourages continuous maintenance of ablution. Some scholars have held that ablution here encompasses both outward and inward purity, and that "believer" refers to one whose faith is complete.