DC Bus for Solar Charge Controller Battery
This is how to calculate how much power you need.  You have two devices: a modem and a laptop.  (0.5A)(12V) + (19V)(3.4A) = 70.6 W.  So you will need to provide at least 71 W of power.

I'm wondering why you want to run the laptop from an external battery.  Does the laptop not already have a battery?

Are you using a UPC that outputs AC 120V?  It sounds like you want to power your devices directly from the charging circuit. 

I would advise against this since the UPC is designed to output the standard 120 VAC.  If you bypass it you will not have any current limiting fuse or circuit breaker in case of a short or overload.