Type 2 clematis are a bit more challenging to grow than type 3 which are bulletproof in my experience. You are correct that they will want a support structure other than the fence so that they are growing in front of the fence. The trellis should have pieces that are not much larger than pencil width in diameter since they grow by twining the leaf stems. Large slatted trellises an inch or more wide tend to be difficult for the clematis to grab.

A few type 2 clematis that I have found relatively easy are Guernsey Cream, a creamy white, and HF Young a mid lavender-blue which varies a lot in how it looks, depending on the light hitting it at different times of the day. Another one, Piilu aka Little Duckling (the translation of the Polish original name) is pink, and is the only one I have ever seen a second bloom on, though it is typically fewer than 5 flowers if any. I never get double blooms here, only single flowers, due to the plant being killed back in winter and losing the flower buds formed the previous year. Flowers on new wood of most doubles tend not to be double (extra petals for additional fullness).

Here my growing season is too short for type 2s to bloom twice. They typically are killed back to the snow line here in central NH, so get pruned back like the type 3s. They bloom in late May while the type 3s bloom in summer. I don’t know whether your area will allow for full double blooms or not.

The longest blooming clematis I have is a type 3, Rooguchi, which has dark purple-blue bells. It doesn’t climb (basically a tall, lax stemmed perennial) and grows to about 6’ in my garden, so I grow it up the middle of a metal obelisk.I get about 6 weeks of July and August bloom on my mature plant.

I agree with all of Bab's comments and would just add that not all type 2's are rebloomers and those that do tend to have a significant gap of no flowers in between the two bloom cycles. And the second bloom cycle is rarely as heavy or profuse as the first.

IME, any of the type 3's will have a longer bloom season than the 2's, although they start later. They are also far less inclined to develop clematis wilt. There are many to choose from but it is hard to beat Jackmanii - dark, purply-blue flowers produced in abundance on a vigorous, mid-sized vine. Huldine would be a good white flowered companion.

My only hesitation for Huldine is that mine bloomed during Japanese beetle season, so it always had tattered, chewed flowers. It doesn’t seem to be an issue with the darker colored blooms like Jackmanii or Viola or Etoile Violette or even the mixed colors like venosa violaceae (purple and white) or the mid-colors like Arabella (non climbing and light blue) or the reds.

Thankfully, Japanese beetles have yet to develop into an issue here on the west coast. I have enough problems with deer, slugs and snails :-)

I picked a good half dozen baby snails off of clematis Taiga yesterday, which has erupted into leaf in the last couple of weeks. I even spied a tiny flower bud developing!

This is my group 2/B Dr. Rupple and it blooms basically all year. I’m zone 7b in central Arkansas. This one is in its second or third season and I need to prune and replace with a larger trellis. I have heavy clay soil, it’s on an East facing wall. It’s my first and I have done nothing to it at all 😁 it’s been awesome and I’m going to buy it a bed fellow to grow with on a larger support.