Is prompting players with alternate moves poor etiquette?
This isn't really an ethics question, it's more that you and Player 2 haven't agreed on which game you're playing.

You want to play a game that involves loose alliances and table talk.
Player 2 wants to play a game with strict solo play and no talk.

There's not much you can do here apart from agree, in advance, which version of the game you're playing.  If you want to play the game differently, you'll need to play with different people or put up with agreeing to something you don't like.  Careful phrasing of this can get you away from this being a right/wrong thing.

What you can do is bring games where open alliances are part of the game.  That could well loosen up player 2 to come round to your way of thinking.  Diplomacy, Cosmic Encounter or Avalon would be a good start.

All this is separate from the issue of people outside the game giving advice.  Personally I find that to be bad form.