For your purposes I would use a rambler, but take care in choosing one, I would not want one that swamped the host plant to the point of bringing it down, especially seeing as it's already dead. Therefore I will put forward one that I grow that I think would suit your purposes but there are a number of others.
An English rambler called 'Malvern Hills'. The flowers are smallish, about 2 inches across and a butter yellow centre, creamy outside fading to a creamy white. They have a light scent. It flowers like most roses profusely from June through July but advantageously it will have a short break from mid July to early August and then it will proceed to flower again, well into Autumn on the ends of new long shoots.
The reasons I love this rose are many, in fact I don't know where to start. The new growth is a gorgeous mahogany red, it has few thorns and hardly any at all towards the ends of the long growths, it's not overpowering, the shoots are long and thin and easily pliable. The leafage is matt green colour and almost evergreen in mild winters, only defoliating when it gets extreme. The foliage has a weeping effect and the plant is clothed to the ground, so no bare stems at the base.
Why I think a plant like this is suitable is, apart from the fantastic main flush when the whole plant is covered in flowers right to the ground, it re-blooms on long shoots it produces after the first flush, and these flowers are right at the ends of these long shoots [rather than in the middle], so if the plant is trained over a tree or bower like structure, then all these flowers should hang down gracefully from it's support giving I think a charming effect for the autumn as well. Mine is against a fence so I have to imagine this would be the effect.
Here is a photo of mine, one of the last flowers taken last Oct. 20th. of course you have to like yellow roses - not everyone does. It grew 6 foot in the first year from a Spring planting, and now it has spread to 20 foot width after 2 years, yet it's not what I would call rampant but quite manageable.
It's readily available from garden centres - pot grown, or it was a couple of years ago.