You have two options, either train or bus. I think train is the best considering the time you have to be there, as well as cost-wise.
#1 train - the train departs SMN station at 5.35am and gets to the Pisa airport by 6.48 - total cost 13.70€
you switch trains in Pisa Centrale and catch the Pisa Mover, which is the shuttle that connects Pisa Centrale to the airport, the ride is about 5 min total.
#2 bus - the only bus right now that takes you into the airport is called Caronna. Only thing is that for you, there is an early bus that arrives at the airport around 4.15-4.20am and ONLY on Tuesdays and Wednesdays there is another bus that arrives either at 5.30am or 6.50am (this would be perfect for you, if you're traveling on a Wednesday). You can see the page where I got the schedule info from:
https://www.caronnatour.com/linea-pisa-aeroporto-firenze-it/ - for bus from Florence, you HAVE to buy online tickets ahead of time.
Cost is also 25€ but of course, the convenience is you are dropped off right at the airport - and the bus has various stops in Florence, so you don't have to pick it up only at SMN. I screenshot the bus stops below the schedule.
Let us know what you ended up using and how you found it, I haven't taken the shuttle or bus in a long time! There were also more bus companies running the route before - and they were cheaper. Who knows, once the economic effects of the pandemic/war wear off and more people travel, we might get back to more companies doing the route. In any case, this is what it is at the moment.
Best,
Lourdes