Way to connect test points on circuit board when both sides of board must be accessible?
Ron's suggestion of pogo boards can be  used. There are cupped head ones that are designed to connect to through-hole pins. Photo from here:

Those particular displays have a duplicate common pin on both top and bottom rows, so you could make a little hand-held tester with 5 pogo pins that is held by hand on the display being tested.

Usually to test a display like that, a test program in the driver is used, which avoids the necessity to add extra connections, tests the driver and tests the PCB for opens at the same time.