Purposefully misinterpreting

To misread is to "judge or interpret (a situation or a person's manner or behavior) incorrectly." and can be applied equally to a written or spoken statement. It is, however, neutral with respect to intent, and to fit your question would be used as willfully misread.

I'm pretty sure you're not using "interpret" in the sense of converting from one language to another. If you are, I believe willfully mistranslate would be used, since "misinterpret" is already widely used in the sense of misunderstanding.

slant

[with object] (often as adjective slanted) Present or view (information) from a particular angle, especially in a biased or unfair way:

I agree that this is ridiculously politically slanted in one direction

or

distort

Give a misleading or false account or impression of:

Many investors now distrust pension accounting because it distorts reported earnings.

or

figurative use of contort

Twist or bend out of the normal shape:

figurative: a contorted version of the truth

All of this imply (malicious) intention.

misconstrue

This is an act of distorting or misrepresenting some other speech or text.

Don't misconstrue what I am saying, you know that's not what I meant.

If you're talking about someone's intentionally misinterpreting in a discussion or debate about ideas, there are specific terms. One common tactic to use a "Straw Man" argument. That's accurate when your opponent answers a question you didn't ask with the intention to dissemble. There's also using "Socratic irony," a trick to bait an opponent to trip him/her up. But neither term is a verb.

Hmmm ... these don't look as good in writing as they did in my head. The problem I'm having is that you're using a verb with an adverb yourself. I realize, though, you can't use a synonym because then you'd be answering your own question.

Misrepresent: "to describe (someone or something) in a false way especially in order to deceive someone : to give someone a false idea about (something or someone)" Merriam-Webster.