What size lumber do I need for a mezzanine floor to support a 20,000 lbs load

This is quickly going to become too complicated to tell you for sure if your deck will be able to support a structure like a pergola, but if it is even half the weight of your typical hot tub then your deck will very likely not support it.

To accurately determine this you need to figure out the additional #'s/sq ft of load that a structure as large as a pergola will place on top of the deck. Doing this is simple with something like a spa where you have a clear square footage area and a relatively even weight distribution when filled with water and people. This is not so easy to determine with a structure where certain points of the structure are going to carry the weight of the structure differently. The weight is not as evenly disbursed.

Right off the bat there I would advise you to consult an engineer with your structure plans to determine if this is feasible and safe. This is completely disregarding the fact that your deck is only as strong as its weakest point. If your house is brick and your ledger board is attached with carriage bolts then even if the joists were on 12" centers then you are still likely putting entirely too much stress on the ledger board. You also could have a lot of problems if any part of your structure is going to sit over that 2' overhang.

In the end, an experienced engineer will probably be able to confirm pretty easily if this will be feasible or not.

Look where the joists meet the ridge beam where you're going to put the door.

Get some dimensional lumber about 4 feet long, at least as thick as the existing beam. Nailing a couple of 2x6's together would probably suffice in the short term.

Give yourself enough room to work, and using a jack post and this 4 foot beam segment, support the joists in the area where you are going to work. Don't over support, as you'll lift the floor above. The idea is that you want to take the weight off the wall where you're working. Jack the post until it's snug, and take out one wall stud in the middle of the door. Check the supported joists for level, and jack up if necessary.

If required, remove another stud, and adjust the jack again.

Put in your door frame and header, and slowly jack the temporary beam down.