to not incur extra cost for the loss of the material PCB Manufacturer faces
I think this is your key assumption which I would challenge. While it is great that you care about other people and nature not to waste the materials, I am afraid that it is simply out of your control in this case.
Each square inch of the board is a cost for PCB house. They strive to minimize losses, they combine boards on the sheets, rotate them, use other tricks.
Since PCB manufacturing is nothing than just a business, what you primary would care about is your - end customer - price for their end product - the PCB. And of course less board size should (usually) lead to less variable cost (the cost which really depends on the production of your board - for example, FR4 material sheet size, amount of mask, etc).
In general, if you are still curious beyond the price as indication of the manufacturer's usage of materials (and their quality :) you always can ask them which equipment they use, how they use it, what they do with your Gerber files etc, but I bet they will try not to disclose much of this info because usually this information is not for your eyes as a customer.
Update: as @BeB00 summarized in the comment
so the correct answer is to just get quotes for different sizes and see
and I would add to it that you must design your board so that device would work as expected, can be mounted into place, and be serviceable (if possible). Trying to stick to some size without real product requirement, from my point of view, is nonsense.
Respectable PCB houses usually advise their clients on the quality of their Gerber designs, and even tell clients if there're any elements which can be changed to (a) decrease production cost/minimize productions issues, and thus (b) decrease total cost for customer.