Use of "that" versus "about"
Solution 1:
That is a conjunction here, it can attach a complete sentence (having a verb in proper tense, etc.) to the previous context.
About is a preposition. You can only attach a noun, a pronoun or an expanded noun phrase using it. To transform a sentence into a noun phrase you need to replace the verb in its tense with a gerund, so that it can be treated like a noun. So 'bite' must become 'biting' in this example.