Were you going to pay the outside hire more money than the internal person you want to switch?

"We'd like her to take on more responsibilities and duties, but we'd like to keep her title and salary the same." - Therein lies your problem. Fix that.

If you were hiring someone from the outside, what would their title and salary be? Offer that to your internal candidate. Anything less is disingenuous and in bad faith.

Simple question - do you (and broadly the company) want this internal startup to succeed or not? Staff it with the best people you have and keep them happy (salary, vacation, espresso machines, whatever). Many times, being on an unsuccessful 'internal startup' leads to being ejected from the company as a 'failure'.

The quitting Engineer was not offered a different title. New hire would also get an Engineer title as they would be a replacement for the Engineer who is quitting.

@JimLarsSvenson - Understood. My suggestion would be to offer the same title (Engineer) and the same salary that you would offer an external candidate. If that isn't acceptable to the internal candidate then don't offer them the position.

@joeqwerty, internal candidate already has the same title (Engineer) as does the one quitting.

@JimLarsSvenson - Understood that you're offering the same title. Thanks.

Also note that if you pass on the internal employee, she might very well consider this a red flag and find another job. I would.

Observation: I think that in the end giving the internal candidate the new title and raise in salary is "cheaper" than keeping her where she is, and bringing someone new from outside. On the first case your costs will increase by the difference in her current salary and the new role salary. On the second case your costs will increase in a whole new full salary to pay... this assumes you don't have to fill the non-senior role she's leaving

"She is interested in taking the role but insists that it is a more senior and more specialized that her current Engineer position describes." - Do you thinks she's right? I ask because this might make me agree with her: "The role is broad, requires self-management, and involves many responsibilities within the startup: engineering, dealing with customers, writing product specs, creating and maintaining new technical and business processes of the startup, supporting other engineers, etc."

@MichaelJ. added an edit about the compensation.

One thing to point out is: the fact that you had someone with Engineer title working this job does not imply that it is a proper tiltle for the role. It may be that the role wasn't getting the recognition it deserves, and internal candidate is really just pointing that out.

Why is the Engineer leaving?

@DKNguyen You nail it! My guess is that by answering this question OP will, likely, provide all the ingredient to answer the original question.