Join Date: Jul 2005
Location: London, ARN, HEL, ..... or MAN
Programs: BA GGL / GFL, Mucci Diamond!, HH Diamond, Radisson Premium, IHG Gold, Hertz Gold
Posts: 5,908
As c-w-s said above, there is unfortunately no chance that your new status will be in place by 5 June.
Even if the flight posts on the same day and your exec club profile shows > 600 TPs, it will take 2 days for your actual status to be amended and visible to the lounge operators and to you in MMB.
You might try to show them your exec club page and they might allow you access, but technically they should not do so until your status is formally updated..