I can't press the R1 button
Could you be a bit more specific?
What is an "R1" button and on what controller (controller usually means microcontroller here)?
What about mounting anything to the "whatever" that is having this button?
The smaller I can make everything the better. The R1 button does not require much EFFORT or PRESSURE so any solenoid should work
Could you put some numbers on that?
Buttons can vary in needed pressure from a few grams to several hundred grams or more and we don't know your frame of reference. Ïf you have no way of obtaining the numeric data, could you perhaps compare it to something generally known?
The numbers needed is initial pressure (in grams/ounces/whatever) and travel (in mm/inch/etc).
but I can't find one that would work with small batteries. They all say 12 volt!
They do come in "all" sizes, but it needs to be able to push the button with the least amount of energy spent if run from a battery.
Another option is to bridge the switch of the button with a transistor, if the circuitry allows it. This will be much leaner on a battery.
You may even be able to install a 3.5mm jack socket on the controller and use the tounge switch to bridge the button directly through that connector.
I intend to use some kind of mouth switch to activate it like this - http://www.conceptusinc.com/tongueswitch.htm
You need at least a transistor, a couple of resistors and a diode in between the switch and the solenoid (if you still want the solenoid), as trying to run the solenoid directly from the tongue switch would wear down the switch contacts in no time, due to inductive kick-back from the solenoid.