If I lived in your climate, I'd try giant Burmese honeysuckle, since it is only hardy in the warmest zones, evergreen, and reportedly has the largest flowers of any honeysuckle. The vine grows much higher than 5 feet, but you could keep it cut back.

There are other honeysuckles that will grow in Zone 10, but many of them will drop their leaves for part of the year. Some other evergreen ones you might want to try are lonicera henryi and periclymenum types. There is a periclymenum type called "Peaches 'N Cream" that looks promising--and compact--though some sources only recommend it for up to zone 8 or 9.

Cutting back to 5' a vine that wants to grow 50-60' is pointless and a waste of time. The work will be constant - most vining honeysuckles are extremely vigorous - and you will be removing nearly any potential for flowering. And fwiw, Lonicera periclymenum is not evergreen but a lovely, big and fragrant vine.

The hardiness zone listings for plants only relate to winter cold - they have no bearing on how hot it could get in summer or how mild it may be in winter. Technically, only a lower or baseline number should be provided, not a range. As long as the required growing conditions are met, it doesn't really make any difference how high the zone is. Any vining honeysuckle can be grown in zone 10 provided it gets proper growing conditions - appropriate soils, adequate water, proper summer humidity, etc. Lack of dormancy could play a factor but zone 10 receives enough days of sufficiently cold (but not necessarily freezing) weather to address this for most plants.

5' is awfully compact for most vining honeysuckles - the common size for most is at least 12-15' and many get considerably larger. I might try something smaller......a clematis perhaps?