Sunour has a workable alternative suggestion.
But, back to the solar question.
You did your calculations for 120 volts when you needed to be on the 12V side.
To produce 79 watts on the 120V side, you are going to be pulling about 90 watts on the 12V side, depending on the specs for your inverter. 90watts/12v=7.5 amps.
So, your 18AH battery will last about 2 hours. You would also be exceeding the 5.4 amp current. (Are you sure that is a discharge current and not a charging current limit ?) If you really need the three hours (and to help prolong the battery life), I would either go with a larger battery or wire two of the UB12180�s in parallel.
Your solar panel is way to small to support your intended usage. The 20 watt rating is a full sun rating, which is for a stationary mounted panel that is pointing south and aligned for the sun angle. In that situation, it would take about 4.5 hours to produce the 90 watts required for one hour of playstation time. Many US locations do not get 4.5 hours of full sun, for parts of the year.
In a flat mount (and mobile) situation, your panel will produce significantly less than the 20 watt rating. My guess is that you have about 1/10th of the PV power that you would need to make the system reliable.
If you already have the inverter, you could do an experiment. With the battery fully charged, hook up the inverter and play station and see how long you can run it. Next day, wire the PV panel directly to the battery and lay the panel flat in the yard. At the end of the day, hook up the inverter and play station, and see how long you can run it.
Inverters that are designed for solar applications generally have a blocking diode or equivalent circuitry in them. If you are using an automotive type inverter, you will definitely need a blocking diode to prevent discharge during non-sun hours.
If you end up linking multiple smaller panels, you should also have by-pass diodes on each panel to prevent them from affecting the systems output as individual panels are shaded or otherwise limited in output.