Ah, I see. It's the inverse of a PC joystick port. Rather than having software charge the cap then measure how long it takes the RC circuit made up of the joystick pot and internal cap to discharge, the software trigger discharges the cap and measures how long it takes to CHARGE.
You can still do the same thing, fortunately, you just have to do it high side. Use a PFET (a PNP will also work, but you won't get all the way to the rail) and again a smallish resistor. Looking at the schematics, this resistor isn't strictly necessary, but it may save your console if you hook things up wrong. Hook up the paddle input (pin 5 or 9) through that resistor and PFET to the power rail (pin 7). Whenever the PWM signal is low, the cap will be allowed to charge through the resistor. When the PWM is high, the cap will not charge. You can, as with my previous post, vary the rate at which the cap charges by varying the PWM duty cycle, though the sense is inverted (higher duty cycles cause slower charge and therefore read as though the pot is at a higher resistance than lower PWM duty cycles). Note that your PWM signal needs a logic level equal to whatever's on pin 7 of the 2600.
There's another way to do it, though. This method avoids all the analog stuff. Rather than use a PWM signal, you can just use some timers and an IO line that you vary between an input and a logic high output. Monitor the status of the paddle "analog" line. Wait for the 2600 to "discharge it" (it goes low). After a certain period of time, which you can vary to adjust what position your fake paddle represents, you perform a hard charge by temporarily driving the line high for a short amount of time - just long enough to charge the cap and trip the input. Then return to a listening state and repeat the process when the 2600 discharges the cap.
You could work out what time delay you need to emulate a given pot resistance mathematically if you determine (experimentally) the threshold at which the 2600 reliably reads a "high" on those inputs. Then work out how long it takes an RC circuit (R=1800+RPot, C=0.068uF - values taken from the 2600 schematic) to charge to that threshold for the RPot of interest.
In this case, your logic high only needs to be high enough that the 2600 to register it.
Either method should work. Which way is easier depends on your definition of easy.