`trivial_regex` false positive and/or incorrect hint for `replace`?

I love the new lints! But I found what seems to be a false positive:

For the following code:

    pub fn from_bigml<S: Into<String>>(name: S) -> Result<FieldName> {
        // Again, we have a fair bit of internal knowledge about what kinds of
        // field names can actually exist. Don't hesitate to add new types.
        lazy_static! {
            static ref PRIVATE_RE: Regex = Regex::new("^private_").unwrap();
        }
        let name = name.into();
        Self::from_unescaped(PRIVATE_RE.replace(&name, "private/"))
    }

...the trivial_regex lint suggests:

warning: trivial regex
  --> src/types/field_name.rs:80:55
   |
80 |             static ref PRIVATE_RE: Regex = Regex::new("^private_").unwrap();
   |                                                       ^^^^^^^^^^^
   |
   = note: #[warn(trivial_regex)] on by default
   = help: consider using consider using `str::starts_with`
   = help: for further information visit https://github.com/Manishearth/rust-clippy/wiki#trivial_regex

Not all regular expressions are used for matches, and I'm not aware of anything in std that implements "replace this string only at the beginning of another string."

Reproduce the warning from the Rust example in src/types/field_name.rs:80, then inspect the trivial_regex lint and how it handles Regex::replace with a ^-anchored expression. Check whether the starts_with hint matches replacement semantics; done means the false positive or incorrect hint is resolved and the reported case has regression coverage.