You should be ok doing exactly what you mentioned. Only thing will be fuel run time. I have a champion 4K on the back of our rig now, but also still have the Honda equivalents (2 - 2000 watt inverter gensets) in my shop previously used. I have the homemade version of the 6 gallon aux tank for the 2 units and can run well over 24 hours with that set up. Without the aux tank, about 4 hours with a 15k AC and everything else going. Not sure how long a 2k unit would run with the smaller AC as you've mentioned. Worth thinking about.
John and Rebecca Dickson
Emma-8 / Little John-5 / Iva-1+
2013 Ford F-350 Lariat FX4, CC LB PSD, DRW
2015 Jayco Jay Flight 28BHBE (#8)