Considering the hourly cost of shop time, I don't think you'll be able to trim 4 pounds from your 8.5 pound rifle for $500, but let look at what might be done. The first element I'd address is the weight of the stock. If you can get by with a shorter LOP, you should shorten it, then regrind the recoil pad to match the smaller butt profile. The next step is to remove wood from the butt area of the stock, which could be accomplished with a simple brace and bit, taking care not to unduly weaken the stock. You might have now shaved as much as a pound from your rifle, without any cost whatsoever if you do the work yourself.
Reassembled, you will probably notice that the piece is now muzzle heavy, so the next step is to have the barrel shortened and crowned. Now you could shorten the rifle incrementally to see where the best balance for you lies, but this would be a slow and expensive proposition, which will probably result in a barrel about 20" long, so just get it cut and crowned at 20".
Consider light weight scope mounts. Traditional Weaver rings are light and inexpensive, but can be a little annoying when attempting to orient the vertical cross hair, however, once the scope is centered and the ring screws tightened to 15-20 inch-pounds, it will never move. The lightest scopes suitable for big game rifles are aluminum tube fixed power models, since they lack the longer ocular tube and a power ring of the variable. The old time advice of choosing a 4X for woods hunting, or a 6X for open country won't leave you wanting in many circumstances. Lighter still, would be to forgo the scope all together and install a ghost ring rear and rectangular post front sight, but today, few are willing to abandon the advantages of modern optics, for 19th Century sighting equipment.
While I prefer slings that provide dual service as shooting sling and carrying straps, I would forgo this when attempting to reduce weight, and choose a simple single strap nylon sling, although I'd probably hold out for an inch and a quarter rather than than a 1" sling. Uncle Mike's style swivels and studs are ubiquitous, and I doubt I wast time looking for a lighter (more expensive) alternative.
Any money you have left in your budget could go towards lightening the bolt. The combination of a skeletonized bolt handle, an aluminum striker shroud, and a titanium firing pin might reduce another ounce or two, but all of these alterations are small steps towards getting a lighter rifle rather than a heavier one. As I said at the outset, I doubt you'll be down to 5.5 pounds, but your rifle will be lighter than when you began.
There is one other option; that would be to sell you current rifle, or trade it in towards a light weight carbine, with those proceeds and the $500 you might get closer to your ideal back packing rifle, than you could by attempting to customize your 700 with a small budget.
)...
