In the 'Parameters' tab, disable the 'Lock Database Schema' setting (for each of the services that access the feature class(es) in the feature dataset). This is on by default, and of course you will need to consider if this may adversely affect anything that consumes the service(s) (some applications could get confused by schema changes).
Of course, you do have to save and restart the service when you change this setting. But that's a one-off restart for each service and enables ongoing multiple schema changes without having to stop the service again.
Documentation for this setting is at: http://enterprise.arcgis.com/en/server/latest/administer/windows/disabling-schema-locking-on-a-map-service.htm
I always disable this setting on all of my map services. It's just too much of a headache to manage even a trivially dynamic GIS schema otherwise. The first thing I did when I started my current job was to disable this setting on all services.
The only other way to avoid this problem is to avoid using feature datasets. Sometimes you need to use them (eg, for geometric networks, etc). But don't get tempted to use them just to organise feature classes into groups.