https://www.houzz.com/discussions/5859544/need-suggestions-for-a-climber-rambler-not-a-modern-rose#n=1

Not sure how large you need, but Teasing Georgia is a favorite of mine. Planted as a band size in 2018 to over 8’ last year. Bloomed all year, although not always profusely.

I've always wanted a place that would work for a Maman Cochet Cl. (or one of the sports). Lady Hillingdon Climbing is available from Roses Unlimited -- though it might bleach out in your Texas sun if it is an all day exposure.

You said you don't mind if it's not remontant . . . Have you considered 'Veilchenblau'? I know it does very well in East TX (from what relatives have told me).

I have and adore Vielchenblau. I first saw it down the street from where a friend lived, it was in full bloom. I asked for cuttings [this was a monster rose that covered the whole side yard] and the lady refused. So I kept at it until I found out what it was. Shame it doesnt bloom all year, ditto Lady Banksia. I have these two planted near each other.