DC backup battery switching module

It seems like there is a delay during switching of the relay.

The relay is a "break-before-make" 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 "make-before-break" -- 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.