The important thing to note about your LCD is not only that it is 16x2, and the pin numbering order, but the controller IC.
Your LCD uses the KS0066u, which, as far as I know is compatible with HD44780 (I am not sure if there are some edge cases where this compatibility is not 100%, but we can most probably assume that you will be fine). The HD44780 is the most common controller for the myriad of 16x2 LCD modules sold on eBay, etc. for use with an Arduino.
Thus, you are lucky that indeed these ubiquitous LCD modules use a controller compatible with yours, and all that need to be resolved is the pin ordering.
My suggestion is that you simply buy a 14-1,16,15 module, and do a little surgery on the instrument's PCB to switch the traces that go to pins 15 and 16 (simply cut the traces and solder patch wires instead). This should be much simpler and faster (and almost certainly cheaper) than trying to hunt down the more rare 14-1,15,16 modules.