E.D.
Hey J.,
Any flowering plant is going to attract a few bees, I'm afraid. Also, you didn't say whether or not your new arbor gets mostly sun or mostly shade. All ivies are shade-loving plants, and will burn in our Texas heat in July and August and die back quite severely with full sun exposure. If your arbor gets mostly shade, I'd personally go with a creeping fig over an ivy. It's a dense, woody vine that will have much better luck in the wintertime growing on an arbor than a delicate ivy.
If your arbor is sunny, you might consider a wisteria vine. Yes, it does bloom, so you will get a few bees in the spring, but the blooms last for less than a month, and the rest of the summer it's a rich, deep green. Also, in the winter months when most of the leaves fall off, the wisteria vines themselves are very pretty to look because of their woody, twisting vines.
Just my two cents, but I hoped I helped! Please feel free to ask me any questions - I've been an avid home gardener all my life, and grew up with a florist as a mother who had a garden that rivaled Martha Stewart's.