Word Choice – ‘Can Hardly Wait’ vs ‘Can’t Hardly Wait’

Per my comment, I'd avoid it because it sounds rather dated and "upper class" to me.

Semantically, the reason for avoiding this construction is simply that it takes the focus off the critical word not. Since the "conclusion" clause is intended to convey something along the lines of "I am failing", this negating word is vital to the sense.

In "injunctive" forms, such as "You must/mustn't do that!", the word must is invariably stressed, to emphasise the intended meaning. In OP's usage, the word must wouldn't normally be stressed, because there's no sense of injunction or stricture (except loosely, in the sense that there's a logically enforced conclusion). It's "not doing it right" that counts, which requires not to be vocalised.

To confirm this particular contraction is nonstandard, note just 3 instances of "I mustn't be doing" in Google Books, but 3830 for "I must not be doing" (almost all for the sense relevant here).

Grammatically, you can use can't instead of can not or cannot in the majority of circumstances. There is an exception. In wh-movement, the contraction should not be expanded unless you also change the word order:

Why can't I have some bacon? //OK
Why cannot I have some bacon? //not OK, archaic
Why can I not have some bacon? //OK again, although formal

Stylistically, the choice between can't and cannot is more complex. Generally, people use can't in speech and informal writing, and cannot or can not in formal writing or very formal speech. Also (as @Kris points out in a comment), cannot might be used when you need to carefully distinguish it from can't in speech.