IME the max height allowed is usually 2m (6' 6''). So yes check with local council in case any local restrictions.
Agree that the combined weight of plants can be heavy, so make sure that all panels etc are in good solid condition before allowing climbers to take over.
A 2cnd trellis actually in front of the existing fenceline could work quite well. Height then up to you as not a boundary. Take care though if over tall & a windy situation. By leaving some space between the 2cnd trellis & fence maintenance of latter will be easier. You can also hinge the trellis to the fence to allow better access. J.