Sadly, there is no current way to change the paramaters of bounce interpolation, but there is a way to get the same effect via functions within drivers that can later be baked into keyframes. First off, you need to understand what mathematical function the 'bounce interpolation' derives from so that the driver can understand what kind of transformative vs time function you want. If you want ordinary bounce that decays exponentially with each bounce you must use the function
f(x) = (e^((2-x)/a)) * abs(c * cos(x * b))
(in this equation I used the constant 'e' arbitrarily, you can input any value as the base) To implement this function to the output of some tranformation (in my example I put translating the Z coordinate) within blender, you must do as follows.
Step 1: first, add a pound symbol to the value you'd like to animate and press enter, this will create the driver
Step 2: right click on the box that you created the driver in and select edit driver, this will open the driver menu. Then type the expression,
pow(e, 2-frame/6) * abs(60 * cos(frame*60)) , in the driver editor expression box. The variable frame is the input variable and is taken from the value of the frame your animation is currently on so the function can be with respect to time.
Step 3: The values 6, 60, and 60 were the parameters I chose for my bounce animation and can be changed to your liking. If you press the space bar to play the animation it will work just like the built in one, but customized, never ending, and procedural!
Step 4: If you need this to be converted into keyframes, just right click on your object, press f3, search and select bake action, and once thats pressed make sure to select clean curves in the pop up. After that is done make sure to delete your driver.
This method works for any function you input into the driver so make sure to get creative and have fun with it!