Type 2 or type 3 doesn't matter as far as climbing ability is concerned. You probably realize you will have to have something on the pole for the clematis to climb on. They won't grow up a pole without support.

I don't have any specific suggestions, just a couple of general things to think about. To help you find a clematis that won't totally overgrow and overwhelm your bird house (such as sweet autumn clematis,) you can search by plant height (in meters) on the search option on Clematis on the Web. Also check the group; Integrifolia group plants don't climb well and so would need help to weave into a support. I would think that regular fussing would be disturbing to the birds, so you probably want to avoid an Integrifolia. As far as support, I have seen others put green coated wire fencing around posts, fastened loosely, for clematis to climb.

My favorite clematis are the summer-blooming group 3s since their bloom time tends to be a when fewer other plants are blooming, and since they bloom on new wood, I have fewer worries about hardiness of the vines and buds (but that is less likely to be a problem for you.) They also don't get wilt, while some of the group 2s seem more vulnerable to that.

Here is a link that might be useful: Clematis on the Web