You can put the MCU and other circuitry on the underside, but perhaps use at least a 3-layer PCB so you can run an internal ground plane under the power components.
Is this keyboard a one-off or are you doing production runs? Try to keep all the SMD components on the same side if you can.
If you're using the pico rather than a bare RP2040, you'll have a much harder time putting anything on the underside though.
