A friend and I have done 30 sessions on a hard and very weedy lake, we have had only one carp out of it in that time. For 29 sessions we marker floated, tried different bait, the method feeder, zig rigs with black foam, PVA stick, Solid PVA bags, different ground baits the list just goes on. On the 30th session with out any real thought I put half a dozen boilies in a spot that doesn't look dissimilar from your last photo, then promtly forgot about it. Sometime the next morning I had quick look and the boilies were gone. Only rod we had to hand was an old float rod set up for sprat bashing, so In it goes with 2 boilies on and 10 minutes later we managed pull out a 19 . So I would suggest one in a clear spot in the margins is a must. As previously mentioned bait up spots in the margin during the week and see if the bait is gone next time your there.
OK, now be the weed, feel the weed, become the weed,,,,,blah, blah, blah, that may sound silly but I do mean it. You have to imagine what it's like down there and how fish will act due to the environment. The way I look at weedy lakes now is it's a road network. Why would a fish put the extra effort in that's needed to swim through the weed if it doesn't need to? Yes there are natural food sources in weed such as snails etc (which should also determine what bait you use) but I suspect they would rather swim in less confined spaces and the trick is to find these highways and intercept the carp. If you do manage to find those highways (routes) then the rig choice does not need to be complicated. A simple KD rig for instance with a critically balanced bait will do the trick, but most importantly is using a safe rig in heavy weed conditions.
If you cant find the routes then there are a couple of options. Create your own clear spot with a weed rake or if you have time get the fish to do the work for you by feeding hemp in the same place on a regualr basis. Or fish a zig rig, find the height of the weed and adjust your zig to sit just on top of it, the weed does a handy job of hiding your line and leader
If there is a lot of natural food source in there which there will be because of the weed then I would fish a hookbait with a natural food source boilie like bloodworm and my main feed (spod or whatever) would be hemp based. Persist and you shall catch!