Frequency about 13,5 MHz and the dimensions prove that there's a reactive load until something lossy appears between the capacitor plates. Obviously (=assuming your generator+the auto-match are designed to work with capacitive load) no matching circuits are needed, only a conductor, as short as possible and about the same size as your output terminal. You need also the ground side connection as good. Be sure that the connections are 1 meter or less or all kind of matching problems start to appear due the inductance and capacitance of the wires. There's no steep limit for them.
I guess the shown plate is closed into a metal chamber to where the body of your transmitter is solidly joined, so the ground side connection is no problem.
Actually you seem to have already a good suggestion.
Obviously you need some insulation around your conductor in certain places. That material must stay solid. Glass and teflon probably are ok.
If your generator happens to demand well matched 50 Ohm resistive load to stay alive and your load is the capacitor with no lossy filling between the plates, you must dissipate the output of the generator somewhere. One possiblity is to design a voltage boosting resonant circuit to get the needed E-field strength with reasonably low actual dissipated power. The simplest form of it is an inductor, a 50 Ohm resistor and your plates, all in series. The inductor must be adjusted to be in resonance with the capacitor just at the operating frequency. Your "auto match" obviously can fix small misadjustments.
At full power you will get 5A current which will generate 6000V between your plates when their distance is 1cm. That's a hefty electric field as volts/meter when the frequency is this high. If you happen to touch the parts, you will remember the smell of your burned flesh quite long time.
With Tesla coil type system you can have much stronger fields. Actually you only need to insert a tap to the coil.The resistive losses in the coil become substantial and they need to be taken into the account when you calculate the extra resistor for right resistive load