You can book the Museum Pass timed entry slot at the Louvre without having the Museum Pass in hand. What some folks have posted in the last few months is that the day they wanted to visit the Louvre with their Museum Pass it appeared to be sold out. However, there were still tickets available for regular timed entry purchases. Since you are so close to departure I'd look at the booking calendar and see if there is still timed entry for passes available on the day you are considering. It looks like they are sold until June 4 at this moment.
https://www.ticketlouvre.fr/louvre/b2c/index.cfm/pmpevent/eventCode/PMP
For this reason, I'd probably check very carefully to see if the Museum Pass is going to be useful to you. It lost some of it's value when they started limiting you to one visit per museum and lost more value when the big museums started going with timed entry after Covid.
The passes are very easy to obtain in Paris, I'd wait. There may also be a Tabac (yes, a tobacco shop) in your neighborhood that sells them too. Look for the Pink decal on the front of the shop.