Based on the photo of your distributor it looks like it is in completely original condition, and that just may be your problem.
 
You're using points, which over time will wear and the gap will reduce. This result in the points not staying open long enough to discharge sufficient spark at higher RPM, so your engine no longer revs out like it use to. Cheap and easy to test. Use feeler gauges or put a dwell meter on it and adjust you points gap according to specifications. 50+/-2 degrees I think. Once your points dwell is correct, your ignition timing will be out so you'll have to get out your timing light and correct this.
 
If after doing this your car once again revs out like it use to, the next step would be replacing you points with electronic ignition. I have not used the Rob Roy Garage kits myself, but they look like a nice bit of kit. Google it.
 
You will probably also notice while driving that your engine may not go so well and then get up and really boogie around 2000 or so RPM, and you may think this is the cam kicking in, but in fact it will be your distributor advancing, finally providing the correctly timed ignition (to a point) because your ignition advance in your distributor has not been set correctly to accommodate the addition of twin Webbers, most notably the removal of vacuum to the vacuum diaphragm on the distributor. Having your distributor re-graphed correctly will give your engine the low down response that you probably aren't aware of.
 
In short, if setting the points dwell allows your engine to rev out again like it use to, I would strongly suggest getting a correctly re-graphed & rebuilt distributor with electronic ignition and you will never ever have to go down this road again.
 
Good luck.

Difficult to tell clearly in the picture, but the rotor button looks 'poor'. Also check the carbon bush in the cap.
Have you checked yet for water in the carbies +/or blocked fuel filter? If it revs cleanly under light load but not full throttle that might give us internet mechanics some clues. If initially revs OK under full throttle, then dies off I'd go with blocked filter.
SFE

Revs cleanly throughout the rev range whilst in neutral

what?
Revs cleanly throughout the rev range whilst in neutral

Free revving produces substantially different combustion parameters than revving an engine under load.
 
I think Chuff has provided a good answer for you.
 
The higher you rev an engine and the better the engine breathes (ie when you have had some head porting done) the more important a good strong, long spark becomes.

it would rev cleanly as there is no load.... 
My 2 cents...  
 
Air... Take the socks off - probably not the problem but for the cost of 4 cable ties, its a quick thing to eliminate
Fuel... Lose that fuel reg (for the moment). I note you still have the side tank - are you aware there is a filter in the base of the tank...Could it be partially blocked. Also how have you plumbed the fuel line in to the tank??? ..I'd recommend you either drop the line at the base of the tank to see how free flowing the tank / pipe is... if the fuel flow seems slow, you could also unbolt the neck from the bottom of the tank to make sure there is no crap there/in the pipe..... If the problem is still there, put a fuel pressure guage somewhere in the fuel line where you can see it whilst your driving,...
Spark. If you can afford it buy a brand new electronic dizzy. If not see if you can borrow one for a couple of hours.
Good luck

Go through electrics 1st as blocked fuel filters/ pump issues typically will result in misses or stumbling at revs. condenser/capacitors, points gap/age or low coil voltage can all prevent coil charging which shows up at higher revs.
Weber Jets have fine holes and could have a bit of crud, check plugs after a run to see if a plug is different colour to others.
May want to consider long term a push on style EFI filter from a falcon or similar. They are plentiful and offer finer filtering than the OEM style plastic filter when running webers.