Intel Atom board (DN2800MT) power supply, can I add battery?
You should be able to replace your existing brick power supply with a 12V lead-acid battery with full-time charger attached.  The problem is finding a suitable charger.

Most car-battery chargers that you can purchase at automotive stores are NOT suitable for long-term use like this.  Either they are not a proper 2-stage or 3-stage charger or they don't have sufficient current output when in float mode.

You mention that your unit needs about 19 Watts.  That's about 1.5 Amps.

Check your local suppliers to see if they have a suitable battery charger.  You want a unit that can be left connected to the battery all the time and be able to supply at least 1.5 Amps continuous while in float mode.

There are certainly suitable chargers available on-line from a huge variety of suppliers.