DC backup battery switching module
It seems like there is a delay during switching of the relay.

The relay is a &quot;break-before-make&quot; type switch. This results in a momentarily cut of the load's supply.

Could this be improved by designing a transistor-based switch?

You have a few options. These two came to my mind:

Have a circuit that is a type of &quot;make-before-break&quot; -- can be done with either mechanical elements (relays) or semiconductors (diode power-ORs or MOSFETs).
Placing a sufficiently large capacitor across the load's supply so that it keeps supplying the load during the momentarily cut -- this can be impractical as the required capacitance can be tens of thousands of micro-Farads.