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.
This system is the reason people stockpile Core Dust
Core Dust Description An unknown substance that can be extracted in small quantities from a Rapture's core. Contains information related to the given Rapture unit's combat experience. Can be used to strengthen Nikkes. boxes and other resources so they can min-max when to level up.