I would suggest levers (e.g. bell cranks) to reduce the 'throw' and increase the available force the solonoid can provide.
However a first step in any solution would be to determine the operating force required for the buttons. Glue a little rubber pad on the end of a bolt, and bluetak the head to the center of the platform of some digital scales of at least 1g resolution. Zero the scales. Hold the phone sideways and press the button down on the end of the bolt, gently increasing the pressure till it operates. Note the weight readout at that point, repeat a few times and average, then convert the result to Newtons force using the conversion factor of 9.8 Newtons per Kg.