Well, it's been a week and the connection has been 100% stable. Thank you so much @tiga05 for all your help!
Only thing is, as Concept is still in beta, for God's sake disable automatic updates. Bastard thing chose to spend 20 minutes updating in the middle of my run on Sunday and I had to pound along in silence like a Neanderthal, my head filled only with the anxious worry that people wouldn't believe I'd actually run as far as I did without it being in Strava. Nobody needs that kind of stress in their life.