A couple thoughts.
You have a bit of a tough task ahead of you IMO. Like you mentioned light weight and low recoil are polar opposites. Also, in some ways light weight and LR shooting are competing concepts too. Thirdly, even if you get a good LR rifle you may not be able to find off the shelf ammo that allows you to get the kind of accuracy you need for the LR shots you are talking about.
I think you are going to have to embrace the concept of a brake. It really is about the only way to reduce recoil like you need to and still have enough performance for elk at LR. And IMO you have to plan around elk even if it is overkill for everything else just to be sure you get the job done on those magnificent and tough creatures!
It sounds like your 7mm is too heavy and kicks too much. Have you thought about having it worked on by a good smith and turn it into exactly what you want? The concern I have for you buying something new is that you don't have any assurance of getting a good shooter and with a limited budget and needing to shoot factory ammo you are already behind a bit there anyway. Sell your 308, and have him make exactly what you want.
As far as cartridge choice and assuming you are going to brake it to help with recoil,
I really think you need a 300 Win Mag type cartridge for your purposes although the 7mm would work too out to 800 yards. If you are going to go the semi custom route you could do some research and choose what factory load you want to shoot and then have your smith throat it for that specific load. This would help in trying to ensure you get the accuracy you need. I personally wouldn't go VLD though as they are too picky with seating depth.
Just one more thought. Think a lot about reducing weight in other areas like your pack or yourself perhaps and keep enough weight in the gun to help with the accuracy you need. Also, think hard about finding a reloading buddy that can work up an accurate load for you then make you a bunch of them. I think you will way ahead if you can go that route.