Airlines rarely weigh carry-ons unless they're too big or look like they're bursting through their seams. In addition to a carry-on, you can carry a "personal item," ie. camera bag, shoulder bag or manpur... satchel
So pack your camera and anything else that fits in the shoulder bag. I generally put all the expensive, hard-to-find locally and heavy gear in carry-on. This is of course not limited to photo gear.
If you have short-haul flights (for example within central Europe), you might be required to check in your carry-on bag at the gate if it wouldn't fit in the overhead compartment. That's the reason why you should always be prepared with a camera bag, and couple of sturdy plastic or canvas bags. Checked bags will
get thrown around, and even if you check it in at the gate you might need to collect it at the belt at destination; more throwing around.
Some use vests, which are definitely great for stuffing some heavy gear into bags. Checking a second bag is generally not very expensive so that's another option - or fly business.
I looked at Pelican and similar cases, but decided against them due to the same reason as you: they weigh too much. I recently bought this
for light stands etc and it's very light and sturdy.
Finally, buy a reliable luggage scale that you can fit in a bag.