Sounds like you have experience but just not with the Winnebago, so that 's good for learning a lot up to now and maybe need a few points.
First, I might mention how many Winnebago there are in each year and type and that leaves a bit in the dark for looking at drawings that are posted online.
I see in your profile you have a 36 foot? But we list 35 and 37 in several different models/floorplans!
Maybe we need to get that cleared as a first step, if you can spot which you have?
Normally I would say, no, the inverter would not need to be on when you have shore power. BUT there is a bit different item when I looked at what I might guess you have, so I'm a bit in doubt, before giving a "real" answer!
RV come in lots of different setups and options, so a first thought might be to ask if this inverter was ordered as an option 40 B or added after?
Do you have any info sheets in the big packet of info you might have gotten with the RV? Big bag should be passed along to new owners!
Often something like a sales page like we might get with a new car listing the options? Option 40B would be a 300watt inverter from builder.
But moving on to other things, running the battery down by using the inverter should not happen when driving, nor when plugged into power as both should be charging the batteries far more than you are pulling out.
Are you familiar with the converter which uses 110AC to make 12 VDC to keep the batteries up? Possibly built into/behind the load center where you have a lot of fuses or breakers? The part of the cabinet that get warm and humms to you!
That should charge the batteries when plugged in. If using the inverter runs batteries down, there is some problem getting 12VDC from converter to batteries for the inverter to take out. .
But also, when driving, there is a solenoid which connects the coach batteries and chassis/start battery as well as the engine alternator together. That works to recharge the coach after we boondock, etc as it puts some power back as we drive. And is certain to be big enough to handle the inverter!
Are you a DIY sort that would like to check on these or what level of info would you like? I can sort this for you to check but I also don't need to bore you to tears if it's going to a shop for somebody else to worry about!
I often hate doing the shop thing because I always wonder if they will know about it either??
Plus I grew up being tight and never changed!!