Scheduling an interview while working full time

So I just had an amazing potential job open up. I've applied for it, and believe based on the job posting that I have the qualifications to at least be offered an interview. My problem is that at my current job I am working full time (day time, monday to friday) and don't know how I can possibly schedule an interview. I have plenty of personal time that I'm owed but because of critical staffing shortages, and the fact that my department is quite busy now (I work in healthcare) I know I will never get approved to use any time off. I don't want to call in sick or do anything like that as I know I will need to ask my manager for a reference (and because I'm not lying).

Any advice on how to approach this situation?

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