A good test is check the battery voltage sitting there disconnected. It is probably still around 11.x or 12 v.
Check it with the E350 running. If voltage is higher it is probably charging fine from the alternator.
Shut off e350. Measure battery. Plug into wall power. If voltage is higher than battery, the converter is likely working too.
I'm with scarpi on this. A DC fridge is a huge draw for a single battery if running from battery alone very many hours. If the battery is quite low, a 2.5 hour drive might not bring it all the way back up depending on some things like the length and size of the wire to the alternator.
Also a brand new battery does not have full capacity until you run it down and up a few times. If all is good, leave it plugged in for a few days, then check that the water in the battery cells is above the plates but don't add so much that it touches the sight rings.
Jim