Your question is confusing, but I'll do the best I can. You've used the word "she" too many times in a dangling fashion (with it being ambiguous as to who "she" refers to). To reframe the question, this appears to be the situation:
You have an engineer (not really an engineer, I'll get to that in a moment) who is leaving the company. You need someone to replace her. You have someone who you think can replace her, but that person recognizes that this new position (the one held by the leaving engineer) is above her salary range, and, if she (the existing engineer) were to try to take this position, she would want a pay raise and title change to match.
That's the situation I'm framing the rest of my answer with, so apologies in advance if my answer is way off base.
Now, the first thing you have to understand is that the person leaving is not an engineer. Her title may be "Engineer", but the way you describe her responsibilities is somewhere between Product Management and Tech Lead. You mentioned 5 responsibilities that she has, only one of which is actually engineering; that makes her not an engineer. So the first thing you need to do is to decide what the role of this person actually is: Are they an engineer, meaning that they do engineering first and foremost, or are they not an engineer, in which case they do some number of these other things, like interfacing with customers, managing technical documents, and mentorship (yes, engineers do mentorship as a matter of course, but when it's laid out as a specific job responsibility it becomes different).
It seems to me that the person you are looking for is either (or some combination of) a Product/Project Manager, a Tech Lead, or a Sales Engineer. Now, you've asked this other engineer, who is actually an engineer, to fill the shoes of those 3 roles. While this second person may be very skilled at their job doing engineering, that doesn't qualify them as a project manager. In fact, it will often disqualify them as a project manager; if they have to interface with clients then they have to have people skills, which is not a quality that engineers are widely attributed to have (to put it lightly, and I say this as an engineer myself). So this person is right to tell you they don't want the job, because you're setting them up to fail (whether you know it or not).
Now, as for not getting qualified applicants: How are you advertising the job? Is the job title "Engineer"? It probably is, given the question. But when you drill down to the job responsibilities, as you've laid out, the job responsibilities are not "engineering" (or they are, in small part). It's no wonder you haven't found qualified candidates; the candidates think they're applying to an engineering job, but they're really applying for all these other things as laid out above. If you're asking them to prep for an engineering interview/role, and then you blindside them by asking them all sorts of other questions, it's no wonder nobody is passing your interview. You probably need to take a serious look at your JD and figure out if it says what it needs to say; it probably doesn't, and you're attracting the wrong people.
Now, as for the final issue, regarding the salary and job title: When you change someone's role, you are changing their role. As a result, they deserve a title change, if only to reflect the work they're doing. When/if this engineer you're putting into this project management role goes to search for another job, if she lists "engineer" on her resume, but then that section is filled with PM responsibilities, someone is going to go "what?", and that's not good for anyone. So, you should decide what title you are giving this new position, and that should simply be the title this person gets. Clearly, as emphasized repeatedly, this position is not engineering, so "engineer" is not a sufficient title; even "senior engineer" is not a sufficient title, simply because the core responsibility of this position is not engineering. You need to figure out what the title is for this position. Furthermore, it sounds like this job requires a lot more work than engineering (as engineering is a subset of this person's responsibilities), meaning that this person probably should be paid more than an engineer, as their responsibilities are a superset of "engineering". So she is right to ask you for a pay raise for taking on additional responsibility.
To summarize: You are wrong on every count: The job you are replacing is not an "engineering" job, so stop looking to fill it with engineers. Your interview is failing because you're attracting the wrong people, specifically engineers to a non-engineering role. And this role has a lot higher responsibilities than simply engineering, so the person taking this role deserves a commensurate title change and salary increase for taking it.