I wouldn't rule out the HTs you mentioned because of zone 5. Many live for years in Montana and Idaho etc.
My mom had a Double delight last 20 years in zone 4! Plenty of sub zero temps. Planted with graft above ground, no less. It died from a very early freeze in october that saw the temps fall from mid fifties to ten degrees. Many trees died too. An abrupt freeze following warm weather is the worst, IMO.
A lot of factors come into play--
if the rose is covered in snow, or is sited in a place that it gets less wind, or if the graft is planted deep, or the rose is own root-- could allow it to easily survive subzero. Especially if it had ample fall water.
Our lowest temp this year was 6 degrees and i barely had any cane die back on Chrysler imperial, stephens big purple, intrigue, josephs coat, don juan etc etc. don juan is very protected by location- but he had zero die back.

You may know this already, i apologize if i am being to teachery!- ...i think when they say a rose is hardy to a certain zone, it means at that zone it does not have cane die back. So a zone 7 climber probably wouldnt climb much in zone 5, but it's be a healthy short rose. A hardy climber though would keep getting bigger and bigger.
Does your rose need to have dead wood pruned off or are all its canes healthy and green wood the next spring?. If you dont have any dead wood to prune, then you have a hardy rose. Otherwise, most hybrid teas can survive zone 5, in my opinion.

The best way to get an ID is to post pictures of the bloom in every stage, the leaves, canes and a full plant photo. It's really hard to ID anything without seeing it.