Yes your fridge will stay cool for a while...If any doubt.....when you know your going to travel...the night before freeze up a milk jug or two with water and leave in your fridge during travels...turns your fridge into an ice box of sorts...to tide your perishables over till you reach shore power...
If there is no real need for 120v during travels....no need for an inverter...yes your coach already has a means of charging batteries (converter / charger)....so it may be viewed as an unnecessary expense if not needed....I take it while in storage you have access to power to keep your batteries charged....or a solar panel for that function ?....Inverters are quite reliable....but issues can occur...I have large magnums in our trawler and our RW....never had an issue....and really like the "smart" charging capabilities...Sounds like maybe you could save a buck and run without...Good luck !
Wes & Denise
2015 38RL, Full Body Paint, Slide in Slide, full Laundry, RV Fridge, 5.5 Onan, Magnum Inverter Charger, Samlex Solar, Trav'ler Sat System. Heated Tanks, R.O. Filtration, Sailun Tires**2016 F-450, Reese 25K Elite