I don't like to sound negative, but there is a common misnomer with ATE that all you do is buy the ATE, plug the UUT in and that is it, but afraid that is simply not true. By the way I spent a couple of years procuring ATE, designing test jigs and writing test programs, and I can tell you that it is no mean task.
Even if you buy an ATE you would still have to build an interface jig and also learn about the ATE, the programming language, and write the actual test.program.
Instead, I would suggest that you build your own test jig which may require some human actions to operate, but will still be much faster than knife-and-fork testing.
I would then recommend that you control the test jig by a processor of some sort: personal computer (PC), Raspberry Pi, Arduino.
But, remember that this is only my opinion. Other members may advise otherwise and they may be able to point you to a suitable simple low cost ATE.
spec
PS: afraid I do not have the time at the moment to research potential equipment for you like I normally would do- sorry