After looking on p55 of the
manual, I second the embedded controller approach. A PIC202 (or an 1840 as has been mentioned) would probably work as would a whole host of others, including an Arduino nano (the cheap clones). There are some advantages and disadvantages to each choice depending, in part, on your programming experience.
First, I would take a look at that switch from the other side. How many connectors are coming out of it? What voltage is there? Not sure if you have already done that. If not, I would advise getting a grasp on what the switch is doing. I have been able to mimic a manual switch press (on, for example, a photographic flash unit) using a micro and an npn transistor (an optoisolator) where the collector and emitter went to the two poles of a switch. That would make things easier, but until you have have poked around the switch, you don't know.
The manual (p55) says to press that button "several times" to get to manual mode. You are saying 4 times and my guess is that is based on experience. Still, the way it is phrased in the manual suggests that there may be timing considerations which are much easy to manipulate in code.
Where is the circuit power coming from and what is the voltage and how much current do you have available? So, if it is 12V and ~100 mA, you could run a 4-wire plug from the base to the pedal (+12, GND, Switch1, switch2). Using promini, for example, you could make use of an onboard regulator. In the case of a PIC202, you would supply the regulator. Similar considerations if you want to use a 9V battery in the pedal.
It is not that I am against a non-micro approach, but at this stage, a micro approach makes more sense...to me anyways.