Here's how I handled my lantern lodge character (who has a GM credit for Way of the Kirin, qualifying him to make this change now, rather than waiting until August 15th):
Add the Sczarni faction, ignoring the error message about too many factions.
On the journal tab, I see he's got 47 fame/6PP, so I add a new prestige journal, name it "Faction Changeover", and set this one to Sczarni. I set "Earn" to 47, and press that button, then set "Spend" to 41, and press that button, leaving me with the same Sczarni fame and PP totals as I have in Lantern Lodge. There will be an error message about the fact that you have two factions, but all your faction traits and vanities won't report errors. From now on, when adding prestige journals, I'll choose Sczarni, rather than Lantern Lodge to assign prestige to.
In August's update (or July's, if I can find the time between now and then), we'll add a button that can auto-generate the new journal entry with the correct fame and PP spending, and set things up so that if you have two factions, and one of them is lantern or shadow, that's legal, but generate an error message if the lantern or shadow factions are added to characters who didn't already have them (and after August 15th, generate an error message requiring a new faction for any lantern/shadow characters who don't have a second faction).