according to folio 22 of this link:
https://dlc.library.columbia.edu/catalog/ldpd:113554

it looks like this would have just been chanted to a formula since the text is from a R/V.

The Neumz app also sets it as a formula: https://app.neumz.com/listen/nona-elegit-eam-deus-et-praeelegit-eam-tonus-simplex-24677/25-03-2021

Of course this is not Gregorian Chant, this is Old Roman Chant. Our master editors Alberto Turco and Frans Kok usually adapt such chants (and also those from the Milanese tradition) to the Gregorian idiom. So here's a version by Frans Kok.

The mode VII melody of Tu es Petrus is a typical melody. You'll find many with the same melodic structure. And yes, I had to think of To es Petrus too because op the solemnity next Tuesday.