Welcome!
If you are going to make just a two-button keyboard with no special features, then I think that the easiest and cheapest thing would be to scavenge the controller from an existing keyboard and connect your key switches to that.
On most cheap "rubber dome" keyboards, the controller is on a small circuit board with the actual switches in a set of plastic membranes. There are those where the controller is on the membrane itself, but they are not as common.
The keys switches of a typical keyboard are logically laid out in the intersections of a "matrix". There are two sets of lines from the controller to the keys: the matrix "rows" and the matrix "columns". This means that each switch is on one line in the rows and one line in the columns. In a rubber dome/membrane keyboard, the top membrane is the columns and the bottom membrane is the rows (or vice versa).
Trace which lines on the matrix correspond to F4 and F12 respectively, and connect the corresponding lines on the controller to your switches. Nothing else should need to be connected. It is possible that the two switches could share the same row or column.