If the GFCI trips at too low of a level it is defective, though it is more on the fail safe side when it comes to the reasons we have GFCI's.
That may be true, but is using a 9 dollar ice cube tester an accurate means of determining what is 'too low'?
Maybe the tester is borderline, working on 'stiffer' but compliant devices, but not on this particular one. I sure wouldn't fail the GFCI based upon info from using a cheap tester that isn't even an approved means of testing.
To me, if it trips with the tester but works with a normal load, the GFCI is just doing its job. It may not be doing it like the directions on the tester say, but it is working.
Those little testers are handy, but shouldn't be trusted for accurate information. If true trip level data is needed, so is a better meter. If there are no nuisance trips and the device works fine when tested with its own button, the problem is likely with the tester, not the GFCI.