Lots of good suggestions above. I will repeat some.
Build the II wing. Save material costs, hangar space, and weight. It will have a higher stall speed.
Cut the spar planks ends with a long taper. (IIRC, and it has been a long time, the specified taper is fairly short.)
The plans specify balsa for the wing tips. Use foam and CF.
Cover the wing with light weight Dacron. Use minimum coatings.
Use fabric to the greatest extent possible. Ailerons, forward fuselage, etc. Use just enough removable covering to allow access to service the brakes and rudder pedals.
Keep fairings light. Use minimum size and thickness of aluminum, or use CF.
Use a CF spinner and a CF cowling.
Use a composite propeller.
Use a small (removable) battery for XPDR and hand held COM. Wire a Battery Tender connector to it.
No alternator, no starter. Use a modified glider tow hook on the tailwheel that you can release from the cockpit. Always have it tied down for propping and entering the airplane. If you absolutely must have a starter, use a lightweight starter and a GPS plug that can be unplugged while sitting strapped into the airplane. One PB to energize the starter.
Extend the longerons to be the four major tubes in the engine mount. That will complicate the firewall installation, but it will save weight. Check alignment three times before welding.
Do add the little bit of weight over stub exhausts by having exhausts that point rearward, not down.
Keep the throttle control as simple (light) as possible. Ditto carb heat. Take alternate air from inside the cowling. No air filters, just a screen a la Pitts.
Keep the fuel tank small. You can have lots of fun with a fuel capacity of one hour with a half hour reserve. Use the lightest off / on fuel valve that you can find.
C-85-8 (if you can find one) with O-200 innards.
Keep the top former of the fuselage aft of the cockpit as minimal as possible. Keep its height as low as possible. Use CF. It does not need to extend down to the upper longeron, it just needs to be big enough to form the curve.
Use a sling seat with a five point harness. You will get dissenting comments on this item, but if light weight truly is your priority, it is an option. (Aerobatics? Go seven point, but then you need an inverted system, etc., so not a minimal weight Cassutt. Note that it will roll so fast that it will not have time to lose oil pressure in a roll, unless you deliberately slow it down.)
Leave the fuselage interior aft of the rear spar open. Do not make a baggage compartment. Note: The first landing will sound like the TW came off, with that open fuselage acting like a megaphone.
Keep the canopy hinges and latches dirt simple and light. Use Plexiglas, not Lexan, for emergency exit considerations.
Use lightweight brakes and wheels. If it is light, slowing down is easy.
Use CF wheel fairings.
Use the smallest aluminum gear that Grove makes, unless you want to land slower, then use the taller gear. I would probably go taller. YMMV.
I don't recall the details of the lower longeron reinforcement for the LG. Probably minimal since it was designed to be a race plane. For a routine flyer, I would refer to the Pitts mods and emulate them, even thought it might add weight.
Use a direct steering rod on one side of the TW, with no spring. Talk to others to get the geometry right. You won't be able to spin it around during taxi, but it works well. Cut off the unused steering horn on the TW.
Use sheer bolts where appropriate, with thin washers and the small nuts mentioned above.
Use small instruments. A panel isn't needed.
No lights, although, depending on the amount of traffic where you fly, you may want to consider adding an LED "strobe" powered by the above mentioned removable battery.
Keep us posted on your decisions and build progress.
BJC