I would like to keep the weight the same or less. I prefer a sporter style stock . Looking for a stiff enough stock to shoot off a bipod and am planning on bedding my action. My rifle is a model 16 7mm-08. I have shot about 100-150 rounds through it and it is not as accurate as a .243 Savage I own with an Accustock. I do not have a set budget but money is a consideration.
Only way you'll keep the weight the same or less is to use the factory stock. There is no replacement stock for Savage rifles that is close to that weight.
Something you can do to reduce weight is to go with the bottom "plastic" and Axis-style mags instead of the bottom metal and magazines with the metal floor plates. This will save you about 7-8 oz. alone.
I have used factory savage stocks for years now, on over a dozen savage rifles, and I have no problem shooting sub-moa groups routinely. These are not bedded either. I see no difference between a bipod or bags and I use both. I think the concern for the flexible forearm is overstated by many. Bedding the front part of the action and around the recoil lug will help, but in my experience not that much. Savages are inherently accurate rifles. I've never owned one that shot worse than 1.5" to be honest, which is why I rely on them. I also have all the tools to tear them apart and mix and match too, so that's another reason.
The only suggestion, aside from the stock you have, would be to look at the wood stock for the Lightweight hunter. You can buy just the stock from Savage, but it will run you about $250. They are very light for a wood stock. I recall about 28-29 oz. I've done this before, for a build I did for a friend. It worked out great and his rifle sits at 7 lbs. even with the leupold scope and talley rings.
IMO the new accufit stocks (with or without the accustock rail) are some of the best fitting stocks I've ever handled. They are light and accurate enough for hunting shots at reasonable ranges. The only thing that might make me want to switch is if I could get a Bansner stock inletted for Savage, but that's about it. Otherwise, I'm happy with the new version of tupperware they put out. It's a big improvement over the old tupperware stocks IMO.