It's not too late to invest the $15.00 in your, by now, many hundreds of dollars router table! However;
I've made a couple dozen cutouts over the years and what works best for me is pretty much what Fred advises. However, I start with a 1/2" pattern bit (has a top bearing) in a router. Most, if not all of the plates I've installed has had a 1/2" radius, so I start out with the required radius in mind.
Using the right-hand rule, I cut plunge thru the top veneer in a clockwise motion with the bearing riding on the edges of the template. The height or thickness of your template depends upon the length of your bit. I use a 1/2" X 1/2" top bearing bit, that's also sold as a dado cleanout bit. After I reach the desired depth, I drill holes in the waste at all 4 corners, and using the edge of the waste as a guide, I cut the waste out. Never had this take over 30 minutes to give you an idea of the job.
Once, I didn't take my own bit to a fellow WWrs shop and the shortest bit he had was a 1" bit. So, we squared up a 2 X 4, and used it as template stock.
Anyway, what ever approach you use....Good luck.