My suggestion would be to approach your local Glass/Glazing shop and give them the exact dimensions of the circle and the hole . Thye will be able to cut this out of some properly tempered or reinforced glass and polish it up for you and make the edges neat . All told it should be less that 50/60 quid and will fit to exactly what you want . Most shops would have offcuts that would be suitable for the blank so reduce the cost as there is not much else they could use them for . Not sure what is around you locally but try a couple of places as with all shops they can differ quite a lot in how they respond , costs and being helpful I found a great one near by and he has helped with a number of glass ideas I had for not much money . In one instance he was so taken with the idea and what he needed to do that he only charged me a nominal sum for his time no material costs.
As in my past life I was in R&D in the glass industry ….any platter of that thickness cannot be cut from tempered/reinforced glass. The tempering or ‘toughening’ process has to be completed after cutting to size.
Standard single pane glazing will be 4mm thick anything lower than that is used fir picture frames and will be very delicate to handle esp if you want a hole drilled in it to.