You would have to read the fishing regulations. Hook and line are OK but using salmon eggs may not be allowed in some drainages. There is a good website - California fish and game.
Why not use flies instead of fish eggs? You could put a little bobber on the line (to help throw it out) and a fly on the end.
I doubt you would catch many fish this way. But, if you find the right location, it may work. Although the fish may seem not to be skittish, they may be quite fussy when a line is thrown in.
By the way, I fly fish, and ALL my fishing gear, rod, reel, knife, case, etc, only weighs 11 oz. I can then almost guarantee I can catch fish. My rod is old- there are lighter ones available now. I use a piece of foam pipe insulation for a case. I do not take a net. I only have the minimum length of line - no need for any more than I am able to cast (which is not that far LOL). A Tenkara system also weighs very little. In my opinion, if I pay the price of the fishing license, I want to be sure I can catch fish. I fish enough to cut food weight back enough ounces to make up for the weight of the fishing gear.
I'm sure others are more knowledgeable on this than me, but I'll throw in my two cents. Most lakes are overrun with brookies. Finding a high country lake with golden's can be be challenging as they don't stock anymore by plane (at least that is my understanding). I've never seen anyone use salmon eggs up high. I've seen people use them in areas that are stocked, but I'm not sure how successful they are. People who are into real minimalist setups typically use fly's. My buddies claimed one of the trout I pulled up on spin bait was a golden, but I am terrible at fish identification. If I can do it, than somebody who has a modicum of skill should be fine (if the lake you are at actually has golden's)
Loc: Colorado High Plains
I don't know the rules for where you're talking about in the Sierras (Golden Trout Wilderness?), but I would guess that there might be special regulations in place, like flies & lures only. Daisy's suggestion of the bubble & fly is good one, you can cover a lot of water with that set-up. Lures work also, but I've found them to be most effective early in the season. If you're talking about fishing moving water, fly fishing would definitely be your best bet.