A few queries...How much maintenance will the fence require? How much time/effort do you have to keep climbers under control?
Purely from my own experience, I've found climbers are the plants that need the most effort to keep them looking good and well behaved (apologies if you know this already).
Are you looking for:
A climber that is self-clinging with suckers or aerial roots, such as an ivy - these can grow rapidly and be difficult to restrain.
Climbers with a twining habit such as jasmine or clematis - these will need wires or trellis to support them. Depending on the variety they can need regular attention to prevent an overgrown tangled mess. The more delicate varieties tend to be the more temperamental ones.
Climbers with a trailing habit such as roses or cotoneaster - these also need a trellis or support and need to be tied in.
Similar to these are shrubs that can be trained as a short climber.
This will also need some support and tying in. The advantage is once established it is likely to need less effort to keep it looking good, is easier to keep within its bounds, and won't grow too vigorously for the space.
Growing two varieties of climber is possible, but allowing them to intertwine will make it difficult to trim back to size and shape without damaging the other one.
My suggestion would be to have several shrubs that can be trained as climbers, and train them early on for fuss-free results later.
Depending on soil type - Thornless (or near thornless) roses such Rosa Gertrude Jekyll, or Rosa Iceberg; interspersed at intervals with Shrubby Winter honeysuckle Lonicera fragrantissima - (not evergreen but winter-flowering with fragrant, cream-white flowers).
Other options for training shrubs: Garrya elliptica James Roof; and Ceanothus thyrsiflorus var. repens. Both are evergreen.