If it were me,
Since you won't have to carry much food or water, you could get away with a very light pack. You'd be surprised. Depending on the shelter you use and your clothing system, I'd say a pack in 30-40 liter range is reasonable (larger for bulky clothing).
And depending on what type of shelter and clothing you're interested in (pricey, because warm clothes weigh a lot and are bulky), 8-20 lbs.
Camping on the beach can really suck, with blowing sand. The lightest, most low key would be a bivy sack (no hassle, just use alone on clear nights) and tarp, but since I don't own or use such a system, someone else would be able to help.
I'd look at a double wall, freestanding shelter with a small footprint for ease of set-up and comfort. Check out the REI Quarterdome (check out sales, think $250 range), MSR Carbon Reflex, Big Agnes SL 1 or 2 (think +$300). Maybe the Tarptent Rainbow, Double Rainbow or Moment ($250), they're single wall, pack up really small but are breezy, so may make beach camping a pain. On second though, Tarptents may be a good bet, they're also low key grey.
For cookware, you should look into the Caldera Ti-Tri system. It burns alcohol, and wood in a windstable way. Get the 900ml-1 litre titanium pot. Costs a lot, but re-sellable ($150). Don't think cooking, but heating water. Build yourself a pot cozy or freezer bag cooking. Washing dishes in the dark makes a mess and takes time.
For clothing, get a set of Driducks Ultralite 2 ($15-20). Full rainsuit and windjacket.
Merino wool baselayers for the anti-stink factor (wear for weeks), and then grid pattern fleece (MEC T3 or Patagonia R1) or any fleece and/or synthetic or down vest/jacket (check out sales after Christmas with Eddie Bauer First Ascent). Nylon pants, board shorts. New Balance Trail runners on sale.
Either go synthetic for sleeping bag (easy care) or low power fill down (5-600 fill, $100). Something durable. Put everything in a trash compactor bag (Ace Hardwear $5) as a waterproof bag liner. No fluorescent colored pack cover.
Mattress pad (R4 plus temp rating), I'd go for an inflatable for packability ($60-150).
Buy used on here, look for sales, you could outfit for $1000 or less.
As for stealth camping, I haven't done it, but the usual idea is set up camp in the dark (get a good head lamp, Petzl $20), and break down before sunrise. Practice strict leave no trace principles .... then who cares, you were never there.
Shave your head (dreadlocks are a heat score) and for God's sack, don't be hanging stuff off your pack (especially flip flops). Clean cut is the way to go, look confused, feign a French accent if you can, and tell them you're from Quebec if you get hassled by the man.