If the engine will run by increasing the mixture on the carbs, that pretty much rules out an electrical/ignition problem.
The vacuum on the carbs will not pull fuel through a dead pump, but the pump could be weak and supplying just not quite enough. Is adequate fuel getting thru your filter (if its full that means fuel is getting into it, but there's no assurance it is exiting at the same rate).
When you hear the fuel pump ticking, does it continue or does it tick for a minute or 2 then stop? The SU fuel pump stops once the float bowls are full and the fuel system is pressurized. If it never stops, that either means the floats are not closing off and flooding the cylinders, the pump is not moving enough fuel to fill the floats, or there is a leak somewhere.
Simple test: Disconnect the line from the fuel filter to the carbs, stick it in a glass bottle or jar. Turn on your key, the pump should tick fairly quickly and fuel should be pumped out of the line into the jar with a good solid flow. If you do get a good solid flow, block the end of the line and there should be a small amount of pressure build up (and the pump should stop ticking or at least drastically slow down if you don't get a perfect seal on the line end). If that happens your pump and fuel delivery are probably OK, and its time to start investigating the carbs.
If you only get a trickle, and/or the pump seems to be ticking slowly with the line fully disconnected and open, start working backwards. Disconnect the line going into the filter, see if you get good flow there. If not, check for a leak or kink in the hard line running down the right hand side of the car under the floor back toward the pump. If all seems OK there, then the pump or the lines from the tank to the pump become suspect.