Salary negotiation email for a job offer

I've had a job offer and I want to negotiate it.

The offer: Senior UX designer, $92,000 base, no bonus, 15 days PTO, hybrid 3 days in the office, start in four weeks.
What I want: I'd like $102,000. If they can't move on salary, a fourth week of PTO or 2 office days instead of 3 would help.
Why: I'm on $95,000 now, so this is a pay cut. I've led two design system projects and they mentioned in the interview that theirs is a mess. Two salary sites I checked put this role locally at roughly $95k to $110k.
The recruiter so far: warm and quick to reply

Write me a reply email. Under 150 words.

Rules:
- Open by saying I'm pleased to get the offer and want to make it work. One sentence, not three.
- Ask for one clear number, not a range. Give the reason that's strongest from what I told you, in plain words.
- If I've given a fallback that isn't salary, mention it only as a second option, after the main ask.
- Don't apologize for asking. Don't threaten to walk away unless I've said I will.
- Don't invent market data or competing offers. Only use what I told you.

Then give me:
1. A short script for the same ask if they call me instead of replying by email.
2. The two most likely replies from them ("that's the top of the band", "we can't move on base") and what I say back to each.
3. Anything in my reasons that's weaker than I think, or that I shouldn't say to an employer.

If what I'm asking for looks out of line with what I've told you, say so rather than writing the email anyway.

What you get: A short negotiation email, and what to say back when they push on it.