From the sound of it, you are carrying only a body with a lens attached and the 300mm. The camera is presumably slung over your shoulder, around your neck, or in-hand, and so you need a way to also manage the 300mm but you don't want to pull out the big camera bag, correct?
The Cotton Carrier system may be a good idea. You would need to carry/wear the vest or belt, obviously, and as you said it would likely get in the way when sitting. I don't know if it would be a concern, but it would also not offer any protection from your surroundings.
I would look at a small bag such as the Nikon CL-M2 that comes with the 300, or the Think Tank Photo Lens Changer 75 as a way to carry this over my shoulder (making it easy to manage when sitting and offering some protection). In fact, that's exactly what I did when carrying a 70-200 f2.8 around, which is about the same size as the 300 f4.
I moved away from that approach, though, and now use a Think Tank Photo Skin Body Bag for "light" travel because it's a little bigger and can hold a lens and flash, and has a few pockets for an extra memory card and battery.
Also, while the 300 f4 may be the biggest and heaviest lens you have, I would encourage you to just use it more to get comfortable with the weight. It's not that heavy and carried comfortably and as part of a larger kit I expect you would quickly get used to it.