Is it possible to repeatedly use a schottky diode at its surge current without damage
The single pulse surge rating is usually a 'maximum stress level' parameter that should only occur under adverse conditions and not in normal operation. If you expect it to occur regularly then the diode's reliability may suffer.

I would use a diode rated for at least double the expected surge current. Heating shouldn't be a problem if it only has to operate once every few seconds. The inductive current will probably decay within a few hundred milliseconds of the solenoid being turned off.

A couple of 60A diodes in parallel should do the job. These are readily available from most suppliers. If you can't find any suitable diodes locally then perhaps you could try scavenging some from old PC power supplies.