Perhaps it won't be had with this set up.
Remember, carburetors are dumber than rocks and they have no brains. You are the one with the brains. Carburetors have no idea what they are bolted to, so don't toss the baby out with the bath water. Look at all the race cars that have won races and set records with Carter AFB carbs on them. So don't admit defeat, before you've really tried to tackle the problem.
I am aware there are a handful of variables in play here, but let's try not to get too technical. At sea level, one gallon of air weighs approximately 2 lbs. 10 oz. At normal 72° temperatures, a typical gallon of gasoline will weigh a tick over 6 lbs. When you step on the pedal and things start happening, Newton's old Law of Inertia comes into play. Objects at rest tend to remain at rest, aye? But since that air is 1/3rd the weight of the gasoline, it is easier to get it moving than it is to get the gasoline moving. Which means, in very rudimentary terms, the air is moving well before the gasoline.
But you have stepped on the pedal, opening the primary throttle blades and exposing the intake plenum to a lot more air than it is prepared to deal with. So the engine suddenly goes lean and stumbles. If one gets lean enough, it will backfire through the carb throat. (Witness all the 'high-tech' tuners who remove the choke blades from their Holley carbs and then try to start their cars on a 45° - 50° morning, for a quick take-off.
Once you open the throttle blades and engine vacuum starts disappearing, you have to toss in some raw gasoline, pretty quickly, in order to keep the intake tract wet. But since that gasoline is heavier than the air, you need to get the gasoline moving with a head-start, so it will be there when the air arrives. In short, advance the pump action and add additional pump volume.
Moved the accelerator pump linkage down one hole (as in closer to the carb body) and then take the car out for a test drive. Better? Worse? Either way, you have a baseline and you have moved off the baseline, so now let the car tell you what it wants. Once you find where the pump rod wants to be, then you can start playing with pump nozzles. But they will be just like the adjustment on the accelerator pump, you are going to have to test-n-tune for your application and your driving style. Use a smaller nozzle and you will have a longer pump shot. On the other hand, using a larger nozzle will shorten the pump shot, by allowing the pump shot to be introduced quicker.
You don't say how you adjusted your
idle mixture screws, but that will have naught to do with the power transition circuit. Those idle mixture screws only control how much air/fuel emulsion is picked up at idle. Once the throttle blades start to open, those idle mixture screws (for the most part) drop out of the equation. Hook up a vacuum gauge and re-adjust the mixture screws to give you the highest possible vacuum reading. It will take some time, moving back and forth between the two. Once you have them adjusted, forget them and focus on the accelerator pump circuit.
If you notice a stumble occurring whilst at a cruise RPM, then it will be time to start playing with step-up springs, to alter the staging of the power mode. But that's meat for a different meal, so let's focus on your acceleration issues, before we start playing with other areas.
