“Just to clarify, I said in an email that I’m looking for $150k to $160k depending on the role. Now that we’ve talked about the job I feel like the $160k is the right amount given my skills and experience.”

That’s how to handle it, with the caveat that you should not be using that as a ploy to get the higher rate, never having had the willingness to take the lower rate. If you give a range, you should expect to be offered at the lower end and you should be happy to accept that offer. If you’re not happy with the lower end then you put forward the wrong numbers.

And if they say no, shrug and move on.