For ultimate accuracy, I think my PCB mill could do this for thin Perspex, certainly for 3mm but probably up to 5mm thick. However, my previous attempts at cutting plastic haven't been that successful. I think it needs a fairly fast router speed to prevent the plastic from melting. I think my machine can run as fast as 180 inches/min but I rarely run it above 24 inches a minute as this is scarily fast when cutting a series of small arcs for example.
The other issue I found was that the plastic can turn into a hard adhesive when it heats and cools and it gums up and ruins the tool really easily.
I remember milling and engraving a tiny steering wheel for a model boat for my neighbour from hard plastic and that took several attempts and a few tools before I got it right for that particular material.
The specs for my T-Tech 7000S mill are similar to those below and it is impressively accurate. Just draw a 140mm circle shape using the Isopro software and then ask the machine to cut it out using the same software.
https://t-tech.com/shop/product/quick-circuit-qc7000/
I think the best bet is the online vendor. Looking at their other stuff they must be very experienced at cutting Perspex! I think I'd ruin several tools on my mill before I got anything clean and accurate.