The issue with simply showing your previous flight's BP is that it isn't necessarily any proof that you're due to receive 140TPs - i.e. OpUps, Redemptions, Avios Upgrades etc.

Obviously the lounge desk might just take a bit of discretion and wave you through as you're obviously set to be a GCH soon, but if they were to follow 'protocol' then maybe only Bobbex's suggestion would work.