The service you are providing is Air Traffic Control.
The Classification of the Airspace governs the separation you are required to provide between flights.
If you can separate those flights that you are required to separate proceduraly then there is no requirement for you to use radar.
As someone said earlier if every thing you are working will remain vertically separated then you don't have to look at the radar or if all the traffic at the same level will be procedurally separated then the same applies.
I believe that having SSR only available does not preclude the ATCO from utilising the methods of identification that one would use when using primary radar i.e. turn method etc.
Therefore when SSR returns are displayed, in addition to the methods of identification used for primary radar the ATCO may use the methods specific to SSR.
However, a big word of caution. If you only have two flights and they happen to be opposite direction one at FL350 and the other at FL360 and you decide not to identify them or validate the mode C (because you have procedural separation), if they have an airprox because pilot A was at the wrong level for thr last 100nm and you did not check, then you will have to answer some hard questions.
If your radar coverage is not complete, ICAO covers this situation with respect to separation standards applied close to the limit of radar cover and against aircraft about to enter radar cover etc etc etc.
In a slightly different way, mixing procedural and radar control is an every day happening even in the London TMA (and every other biusy terminal airspace in the world). The place could not cope with the large amount of traffic unless they used it.