There are only three options to unlock more slots on your synchro device:
If you want to get your units past their level cap (SR at 160, or SSR up to 200), you can place them in the Synchro Device, and they will automatically get to the current max level of your top five characters.
This means you only need to focus on getting 5 units to MLB (3 dupes for level 200) and as a F2P player you can focus on getting only 1 copy of the characters you like afterward.
After level 200, you will pay resources to level everyone in the synchro device directly. Cost goes up as you progress further into the game, so don’t worry if your progression gets slower, since that’s the same for all idle games.