For a rectangular hole, consider using a fairly small diameter drill bit, 3 mm diameter or smaller, one ideally designed for drilling acrylic. Too much pressure may crack the plastic, hence the reference to a plastic cutting drill. There have been posts suggesting a conventional drill run in reverse will work, as well as using a masonry bit (not in reverse). Smaller is better.
Create a template outlining your rectangle and drill the initial holes in the corners.
One may be able to create suitable results from this point by using an oscillating tool with a fine tooth (high tooth count) cutting blade or an abrasive blade.
One can also repeat the small hole drilling along the line of the template. Once enough holes are drilled, application of force will snap free the inner rectangle, leaving a series of protrusions which will have to be filed or sanded flat.
The oscillating tool will be challenging to control compared to the drilling but will result in less post-processing overall if one can manage to properly direct the blade. Consider to apply multiple layers of masking tape outside the rectangle outline to protect the uncut area. If you have the material available, use metal strips to create the outline, providing much stronger protection and also a guide for the cutting blade.
Once you have the rectangle removed and the surface finish to your liking, you can play a butane torch or heat gun along the exposed edges to create a smoother surface. Use caution as overheating is easily accomplished as well as heat applied outside your desired area.