Ask the professor for advice rather than a rec letter.
They're going to be more familiar with the hiring/admissions process than any of us here are. The situation is rather simple to explain, as you've done here: one of the best-positioned people to write you a letter of recommendation is the very person you are applying to work with.
In cases where it would be appropriate for them to write the letter, they can offer to write it. That may make the most sense when the applications are judged by some other committee. In case the primary decision making is really up to the professor (and perhaps their colleague) and the letters are more of a formality, they can say this and you can collect some other less-than-stellar letters without worrying too much, or maybe they'd still write the letter just so it's on file and can be referred to if there is ever a question about hiring practices at the institution. Or, it may be that they ask that you have outside letters and then write their own additional letter as sort of a "friend of the committee" notice offering to fund/admit you to their supervision.
For what it's worth, when I applied to PhD programs (in the US) I also applied to my undergraduate institution (though I ended up going elsewhere). My letter writers included the chair of admissions for that graduate program, and another professor there - I can't see who else I would have asked, and this didn't raise any concerns, but also this is the sort of program where students are admitted to an overall program rather than directly to a lab. Occasionally in my field, students are sometimes admitted directly to a lab when a professor has funding for that student. In that case, I'd still expect them to write a letter, but it wouldn't necessarily be a recommendation letter per se, instead it would be addressed to the admissions committee basically saying "if you admit this student, I'll pay for them."