I've cut thinner sheet goods on the lathe using two sided tape, a round wood block in the chuck, and a blade in the tool post. The last time I had to size smaller holes I made a hollow punch and a carrier for the blank. The punch went in the tailstock chuck, slowly advanced it with the blank turning. Went well..
At the tire factory they machine rubber at room temperature by grinding. The "cutters" are like the carbide grit edge type, looks a metal grinding/sanding wheel. Try an abrasive type bit in a high speed spindle, like a Dremel tool or similar. The finish of a white wall and white letter tires is one of the results of this type of grinding. You do have to control the feed, perhaps some air to get the rubber dust clear. Our white sidewall buffers blow air while the grinders are down.
I have successfully machined rubber on the lathe by grinding with the tool post grinder. That's about the only thing that has worked reliably for me. For your application, how about putting a small grinding point in the Dremel?
That's an awfully small hole. If it was any larger, I would make a hollow punch, like a leather or gasket punch,punch, and press it through the rubber.
------------------------------------------------- Here's something to consider.
Depending on the hardness of the rubber, it doesn't respond well to straight punching, but does quite well if you rotate the punch. To hold hole size, the punch should have a body size of the desired hole, with a hole bored, maybe ten thou wall thickness remaining. The cutting edge should be beveled towards the bore in this case, so the OD is the determining diameter. If the piece that is removed from the punch is the target piece, then the bevel should be the opposite direction, so the bore determines the size of the part. In that case, the bore should be the desired size, with the ten thou wall added to the OD. The cutting edge should be sharpened by spinning the cutter at high speed, applying fine stones to both the angle and the straight sides, alternately,until the wire edge has been eliminated. The cutting edge should be very sharp.
In use, spinning at a slow speed is adequate, and a little lubrication often helps. Depending on the rubber in question, you may use a light lubricant (kerosene) with success, assuming the material is oil resistant.
Cutting a clean hole is best accomplished by cutting into a piece of wood, or other soft material that won't readily dull the edge. I've used a piece of urethane with excellent results. Do a dry run with a stop set, so you barely break the surface of the pad on which you cut the rubber. The pad is best clamped in place, so the punch always hits the same spot. That way you don't damage the cutting edge by repeatedly hitting the bottom of the formed circle, and you get a clean cut. This work is best accomplished in a drop spindle mill (like a BP), but you can achieve reasonable success with a drill press, assuming you can slow down the spindle adequately.