They do make "instruments" for accurately measuring such hole diameters; they are called digital calipers (or veneer calipers if you like the analog type). You can find them at Harbor Freight for about $20.00. Not super accurate, but for this application it`s "close enough".
By the way, the "centering dowel" is a great idea for concentrically locating a foam pad. You may already have a "dowel" in your tool box if you have some deep sockets or even regular sockets if you do not need one that is so long, depending on the thickness of the pad you are trying to center. Just find one that is the same diameter as the hole in the backing plate. Plus, if they need to fit over a screw or bolt head on the backing plate, they are already "hollowed out" from the hex inside shape to fit over the screw and act as a clearance recess.
Your centering dowel idea has been used by mechanics and assembly personnel before when trying to line up larger parts or parts the require accurate location by using long set screws with a hex socket drive (think Allen wrench) to screw into the mating part, and then slide the other part being assembled part over them, attach a few screws or bolts, and then remove the set screw(s) and finish screwing in the remaining bolts/screws where the set screw(s) were. Works like a charm.
For more accurate locations, there where holes drilled/reamed into parts for locating pins. Locating pins or dowels had two diameters: one larger end for going into the stationary part, and one slightly smaller end (0.003 to 0.010 inch, depending on the accuracy needed for location) for the mating part to be removed so it could slide or fit over it. Once the two parts were accurately located, the assembly personnel would use the removable part with the hole as a "template" or guide to drill a new hole in the stationary mating part, ream BOTH holes for the locating pin with a larger diameter end (The ream was slightly smaller in diameter than the large end, usually 0.0005 to 0.0015 inch), and then drive in a locating pin with a punch as a press fit so it would stay in. This way when the parts are disassembled in the future for maintenance or repair work, they could always be accurately re-assembled in their original positions to each other without fussing. A lot of extra assembly time was required for this, but it made life MUCH easier for factory maintenance or repair mechanics in the field working on such production machinery. More-than-you-really-wanted-to-know (MTYRWTK).