Hello, and welcome to CAG!
I don't have much experience fishing lakes, but I fish large rivers frequently so I'll try my best to help you out!
Firstly you mention that there isn't any weed growth in the deeper water, and that is important to take into account. You want to fish as deep as possible to avoid being swarmed by perch, but also shallow enough so there IS weed growth. You will likely only find clams, crayfish etc. in shallower areas with weed growth due to the amount of sunlight. Crustaceans feed on Phytoplankton, which require sunlight like algae does. If there is no sunlight at depth of 40ft+, no Phytoplankton or Crustaceans will be there, and no carp as well! You may find carp temporarily passing through deeper water, but they are likely to feed only in areas with ample weed growth, and therefore food.
Your best plan of attack, in my opinion, is to find the exact point where the weeds stop growing. If swimming is allowed and the water is clear enough, you could find it yourself with a good pair of goggles. Alternatively, you could cast a lead out and feel around the bottom. You'll be deep enough to avoid weed growth, but shallow enough so that feeding carp will still be able to see your hookbait. Bait that spot every day, at the same time of day for several weeks. When you want to fish, it will revolve around your 'baiting hour' as I call it. So if you've been baiting at 6:00 PM every day, you could fish from 5:00 - 7:00, the period of time when the fish will be most active.
Perch are going to be your biggest issue, so you will want to avoid chumming with sweetcorn, birdseed, or any other small particle. You want to keep your prebaiting limited to boiled maize (deer corn) and boilies, baits that are too large for the small perch to pick at.