So, in 1.3, the powerup state is different for each character. This is reflected when you select a character to start the game, as well as when you change characters from the pause menu.

In SMBX2, this remains true for when you start the game. So if mario has the fire flower and toad has the ice flower, starting as mario will give you fire mario, and starting as toad will give you the ice flower.

But then, if you change characters at the pause menu, the powerup state of the character you're playing as will carry over to the other character you choose. This means that if you chose toad at start and have the ice flower, then switch to mario, you'll still have the ice flower. It gets even better, though. If you now beat a level as mario and still have the ice flower, not mario's powerup state has been changed to ice flower at start. You can essentially duplicate powerups for characters this way when you start the game.

after disovering that you can change the startup powerup state, I'm using this to play as Raccoon Toad and Mario in The Invasion 2. However, Mario also has the hammer suit in reserve for when I need to take down bosses. I'll be using that for the first two (perhaps 3? haven't gotten that far yet) invasion levels, as well as rawest forest and the eventual bowser fight.

If needed, I can post a video of what I described.

I would like to point out that this is a bug in Beta 3 of SMBX 2 and it has been stated that it might be fixed by the time of Beta 4.