This is not the sort of thing I have any expertise with, but I like taking a stab at builds, so I'll hazard a guess.
There are a lot of variables at play here. Primarily, the choice of string will be a major factor. Most 'string' will have built-in elasticity, and in this particular application, the string choice will definitely affect the design parameters.
You ultimate goal is to move the wedge down an amount that is equal to the height of the wedge, and you want to achieve this by deflecting the string at some angle which will cause the string to be pulled a distance equal to the wedge height.
If we assume that there is no string elasticity (which is an incorrect assumption), The parameters we have to play with are the overall length of string, the location of the deflection fulcrum and the height of the wedge. Below is an example in the variability of how wedge height and fulcrum location will affect the deflection angle.


Some inherent complications of your design appear to be that you do not have a mechanism to convert the direction of the string pull from lateral to vertical. This will inevitably invite fluctuations in the deflection necessary and will also increase stress in the design, which will lead to failures and binding, all of which will affect its functionality.
I would try to add some sort of pulley-style thing to convert the pull of the string to the right direction.
Another issue i see is that the fulcrum point is not steady it looks like you are relying on the flexibility of the body for form and fit, which means you will not be able to control the fulcrum point, which means you have to design to meet all the fulcrum point locations (within reason).
So now to address the string. If you need to use string, i would find something that doesn't have much elasticity... maybe something like the cables used in bikes. That said, I think the old trope "you can't push a rope" might be a good thing to consider here. Is a string the right tool for the job? Could you use the same material the body is made of and create a 'ribbon' which will not have as much tendency to stretch?
You could also switch to an electromechanical mechanism (like a solenoid) to release the web. You can use the two states of your button. You can have a small circuit that while the button is pressed the solinoid is activated and activates the solenoid, and then once you release the button, it starts retracting the web.
CADnoob
