What about some strategic thinking here rather than antagonising young, culture hungry licence payers.
Close 6 Music down in traditional broadcast format--but maybe share the output (overnight-if necessary) with Radio 3 or just have it available online only.
All 6 Music listeners are technically hip and savvy enough to find the broadcast on whatever platform it is. Outsource it-if necessary to Spotify or Itunes.
If the BBC are trying to encourage multi-platform output, save money and be original, then this idea does two three things at once.