Hi Luca.
A few clarifications so others reading will not be confused.
First, the Oly E-MII is not a DSLR – it is a mirrorless camera and lighter than many DSLR's.
Second, I assume you have the Olympus 100-400 mm lens (as opposed to the Panasonic-Leica100-400 mm lens which would also fit your camera). According to the weights listed at B&H photo, your camera and the Olympus lens should weigh under 2 kg –1694 g to be precise. The Panasonic-Leica would weigh 135 g less. Your camera, together with the panasonic lens, is one of the lightest birding setups with that amount of reach that you can get outside of a superzoom. Personally, I don't have any problem carrying the panasonic lens with the Olympus camera all day – though occasionally my shoulder becomes a little fatigued, and then I just switch shoulders. How are you carrying your camera? If you are carrying it around your neck along with your bins, then I can see that that would be too much. Have you considered switching to carrying it on your shoulder, or trying a sling strap, etc.?
Finally, yes, if you're only interested in bird ID rather than image quality, I believe one of the more powerful superzooms would get you what you need. Though you would have a much smaller sensor, so less ability to crop. But I should add I haven't used a superzoom for many years.
Hope this helps,
Jim
Many thanks! Yes, I have two lenses for my Olympus:
The Olympus 100-400 is fantastic, focuses very fast. Yes, the camera with the lens is less than 2Kg, but add 100g of strap, and you are very close to it.
The lens is big enough that it's cumbersome to hike with the camera bandolier style, even though I have one of those nice Peak Design straps. It would be tolerable if I had only the camera, but with also my 8x42 binoculars it's a bit much. So I generally carry the camera in a backpack, and when I need to take a photo for ID, I need to take the camera out and use it. This is not too bad, but sometimes I carry also a scope ... and it can get really difficult. Also, I like to go light sometimes; if I have to start fitting in the backpack food, water, the Oly with big lens, wind/rain jacket, etc, it gets a bit much.
I also have an Olympus 75-300 lens, which has great quality, it's just a bit shorter and a bit slower to focus than the 100-400, but much lighter. With that lens, it's not too much of a bother to carry both camera and binoculars out, and the camera fits much easier in a backpack that also contains other things.
But for when I want to go light, and just have a camera to disambiguate difficult IDs, the Canon SX70, at 500g, has real appeal.
More so likely than the P950, as the P950 is the same weight as the Oly + 75-300mm lens, and I am not sure which is best (I am also super used to the Oly, which I can use in a completely instinct-based way).
I bird enough that having a light camera for my walks would be a reasonable investment, if it worked as good or better than my Oly + 75-300 for ID.
Yes, Oly cameras for nature photography are simply fantastic due to their light weight, ruggedness, and quality, and when I am after top quality photos, rather than ID photos, I take the Oly + 100-400 and the results are simply outstanding.