This is perfectly doable. However the power supply used must not exceed the open circuit voltage of the PV panel that's connected in parallel (to prevent backfeeding into that), it must be able to withstand the PV panel open circuit voltage being backfed into the power supply, and it must either be able to handle 13A of load, or handle overload gracefully by entering a constant current mode rather than shutting down. The first two constraints could be resolved by adding diodes (or ideal diode circuits for higher efficiency), the latter is an inherent property of the power supply. A 100W power supply also might just be a bit too weak to charge up the batteries quickly enough for your needs. A 24V 15A or 48V 10A power supply might be more appropriate to get the battery charged quickly.
