The need to reduce Inverter noise is very user-specific for sure. If it is used for something like a Hairdryer only, you'd never hear it if it were 6" from your ear
Microwave would be same kind of situation. In your case, not a problem with the envisaged use
I am mindful of inverter noises as I use (and specify them) for use with 240V fridges, where the inverter can come on at any time day or night and the 'wrong' inverter can be annoying.
It is similar - and maybe more relevant - with chargers which are likely to be running overnight if on site with EHU. I have replaced Chargers that have worked fine with different chargers in the past for people, the issue being the charger had a fan which always ran - so on campsite was just too annoying to live with.
I only made my suggestion when I read that you had got rid of the overhead storage area in the cab where in my Renault Master (medium roof) I made it into a proper storage lidded area where I can store all our walking coats 6 in total, all the dog coats 4 in total, a king size throw over fleace and numerous other squashable items.
Overcab storage is a tricky one
My VW LT, like the Sprinters and Crafters, didn't have any kind of overhead storage area as standard (unlike the Masters, Transits and others). This is nice in one way as it keeps the whole cab area open and it remains part of the overall camper as the ceiling line is a continuation.
However the downside is that there is potentially a lot of space that could be utilised which is wasted? I tried experimenting with the LT and put a "soft shelf" above the cab where I would have maybe liked a storage shelf just to see how I could live with it ... I found I kept hitting the shelf with my head as I moved from cab to rear (in that camper build, the cab seats were the main seating area FWIW). I decided to cancel the thoughts of having a shelf there!
On my current Motorhome, there is an overcab bed and on this one I am finding I usually don't bother to slide the bed back and leave it in "sleep mode", which does cut down the headroom in the cab a fair bit (so the opposite of the LT in fact). But I have got used to it and it doesn't bother me - but I think a key difference is that although the cab seats swivel around I don't tend to use them and use the side benches instead. If I were getting into and out of the cab seats when parked up, that low roof could be quite an annoyance.
Conclusion on whether to fit a shelf or not? I have no conclusion