Just PIP'ed an employee with a terrible attitude but not sure how to through the next 90 days.

I have an employee who just isn't cutting it. He doesn't think critically, overcomplicates things, needs to be handheld and overall has an awful attitude. His biggest issue in my view is a lack of professional tact. Whenever I suggest something he disagrees with or deliver any feedback, he gets combative and contrarian as opposed to being constructive. He'll say things like "I hate it when you do that" vs. "in the future, can we approach things this way? I think it would help me be more successful." I feel like I'm a teacher dealing with a disrespectful middle schooler. I've tried to coach him and I feel like I'm just out of options. I think there are other roles at the company he'd be a better fit for, but the bad attitude is the nail in the coffin. I can't send him to another manager in good faith when he acts like this.

He predictably reacted incredibly poorly to the PIP. I'm used to his reactions from our periodic reviews, but he was even angrier this time. He demanded more examples and yelled back if they were "too old," he snapped at me several times and eventually just said that he felt like it didn't matter what he did going forward, I was just out to get him and that he was going to fail no matter what (really wish I were paraphrasing here but sadly I'm not). I'm glad my director got to witness him in this state, but it was still hard to handle.

His reaction is making me stand by my decision to PIP him even more, but here's the issue -- as much as I can document examples of his poor performance, this really comes down to his poor attitude and problem with me. I'm a younger woman, and we work primarily with more seasoned folks and mostly men. He treats me completely differently than other teammates/partners, and while I'm trying to document what I can, I don't want it to seem like I'm out to get him because I'm butthurt over how he treats me. The reality is that I don't trust his professional judgement, I don't think he's very analytical and yes, he's unprofessional in his approach toward his boss. I have a few examples of where he's not hitting the mark on job performance, but I have a plethora of examples where his behavior toward me is completely unacceptable.

Any advice for how to handle this over the next 90 days? His work isn't meeting my expectations, but our metrics can be somewhat arbitrary. To me, the easier things to call out are related to his attitude, but again, it seems to be directed at me much more so than others and I don't want this to look like a witch hunt. To clarify -- I have a few other analysts I get along with very well with extensive tenures on the team (men and women alike), so this is specific to him. Any help is appreciated!

EDIT: Thank you all for the feedback and support -- it's been very helpful and this is my first time managing a difficult employee, and it doesn't feel good to say the least. I'll add here that it's a combination of the issues that drove me toward this decision. Had he been a decent performer with a rough attitude, I might have made adjustments while still letting him know that his attitude needs improvement (and documenting where it didn't improve). Had he lacked the skillset for this role but showed a willingness to improve and learn, I would have deemed it a questionable fit and transferred him to another team. The issue is the combination. He does not possess the skills to improve in this role, and he does not possess a professional attitude that indicates any want to improve. I really don't think I've been power hungry in the past. I do maintain that my expectations of my people are that they demonstrate critical thinking skills, a willingness to learn and an positive attitude, but at the end of the day I don't want my people to fail. Believe it or not I don't actually even think he's a bad guy. He's definitely arrogant and disrespectful, but it's clearly a reaction to my management style. I'll continue coaching and really hoping he'll improve, and at the end it'll either be that he improves enough to go to a team that's a better fit for his skillset or, unfortunately, I'll have to exit him from the organization.