I don't know of any clematis that flwoers that long. Ones in flower in April tend to be over by May/June so I suggest you go for one like Alionushka which will flower from June to September and maybe a bit later in a good summer. http://www.clematis.hull.ac.uk/new-clemdetail.cfm?dbkey=1 It is a group 3 so needs cutting back hard every spring and, as with any clematis, a good feed to encourage all the new stems. Group 3s tend to be the easiest in terms of pruning care. However, what you eventually choose will depend on how big you want it to grow and what colour as well as what flower form. You can do more searches on the site I have indicated.
Helo, I think you have a stunning garden with many beautiful features. How about the Clematis Taiga, gorgeous two tone flowers from June to Sept, it's in the group 3.