Yes, you can put your release live on streaming stores on the same date as an 'Exclusive Store Date' by using the 'Schedule' tool (as the exclusivity period only applies to download stores).
Many stores prefer the release to be live on streaming platforms during their store exclusive period, as it allows them to promote the track through their playlists on the streaming services.
If you do this, it's important that you do not set a release date on any other download stores during the download store exclusivity period (e.g. if you set a Beatport 2 Week Exclusive, do not set the release date on any other download store until after the exclusive period on Beatport).