[Small Fry]

My games play with the rule your describing (dead button rule) but when it comes down to heads up I forego the rule and both players will post blinds. So in this case the same player would post the SB twice in a row.

Lets look at your scenario over several hands if we continued with the dead button rule. Hand 1 (which you described would have player 1 on BB and player 2 post nothing with the button. Player 2 acts first preflop second post. Where would the button end up on the next hand? It would be dead as you're still playing through that eliminated player. Player 1 posts SB, Player 2 posts BB and button is dead at player 3. Player 1 would need to act first on this hand both preflop and postflop . The next hand player 1 gets button, player 2 posts BB. Now Player 1 would post SB.

I think it's best to have both players post and the only thing to keep in mind is that a player cannot be the BB twice in a row.

Think about what would happen if BB eliminated button. BB cannot be BB again so he gets to be SB. Makes sense but heads up he's also the button. But what about SB, who would have been button next? He not only doesn't get button but has to post BB.