No, not unless you are running a lithium chemistry battery that has a higher resting voltage. The Dynon system will not charge the screen backup battery unless the input voltage to Skyview is above 12.25 volts. You could certainly put a battery charger on the main battery and power up the system as long as you like. Section 11 of Skyview Users Guide revision K.
To conserve your aircraft battery, the SV‐BAT‐320 is only charged when the SkyView HDX
display detects that your alternator or generator is online. Specifically:
• If SkyView system voltage is above 12.25V: SkyView runs on Avionics power and
charges the SV-BAT-320.
• If SkyView system voltage is above 10V, but below 12.25V: SkyView runs on Avionics
power but does not charge the SV-BAT-320.
• If SkyView system voltage is below 10V: SkyView runs on the SV-BAT-320. This
discharges the SV-BAT-320.
Although Dynons documentation says it may take as long as 4 hours to charge a fully discharged backup battery, in my experience it's always been more like an hour and a half - do you know your backup battery is healthy? Has it passed a battery test lately?