I'm glad it is working. [If you are a perfectionist, experiment with submissions like -1/((x^(1/3))^4), -(1/x^(4/3)), and 1/(-x^(4/3)). Unsure what to expect, but see if anything makes you unhappy.]
One thing: your custom checker should set bizarroPow to 0 at the end, not to 1. (Copy-paste error no doubt.) In this example here it is irrelevant. But if you use this as a template and have code that follows the answer checker, you would want that flag to have gone back to 0.
Next thing: I removed all the bizarroPow from your example (two places in the checker, and the two lines in the preamble), and it rejected -x^(-4/3) as "Your answer is equivalent, but please use the proper format". So either our systems are behaving differently (quite possible if we are on different branches) or something else was off with what you tried before.
Last thing: I'm confused why with both bizarroDiv and bizarroPow in play, that it rejects -x^(-4/3). But I confirm seeing that same behavior. If you try again and things work out without using bizarroPow, my recommendation would be to remove it. Only because if I investigate and there is something about bizarro that is inconsistent with how it is supposed to behave, I may "fix" that one day, inadvertently breaking your problems.