Should I tell my current employer about a job interview with the company they contract with? A bit of an odd situation.

I am currently a public health supervisor/lead under contract for a managed services/recruiting company. They are contracted with an organization (who provides the work and is their "client"; government entity). I was told by an employee at the organization to look into a new job posting for a PM/training coordinator role they had as I may be a good candidate for it. I have been offered an interview and it does coincide with my work hours. So of course, I will need to take that morning/day off to attend. I have a good relationship with my current supervisors at the managed services company (such as with my ops manager, program manager, etc) in that they would provide me with a good referral, I have always delivered, etc. This other position I am interviewing for would involve me working in a different capacity but on the same project and I would likely be interacting with current peers at the recruiting company still.

Here are a few things I've thought about with this:

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• Not telling them about the interview can look a bit shady
• If I tell them and I dont get the position, I don't want to seem like a "flight risk" since they would now know I am applying for other roles. I actually am not the happiest there and have been studying for an unrelated cert to look for other work anyway, but I'd like to walk away on good terms.
• In my new role, I would still be working with my peers and supervisors/management
• The interview is scheduled in two weeks from now (never had one scheduled that far out), I'm not sure if they will want someone to start right away or if they would be fine giving me a few weeks to provide notice to my current employer. But again, they all work closely together as their client -- in which case, it's possible they would tell my contracting company that it would be sooner that they want me in this role.
• A bit of a side message: I was hoping to take like a week off in between, but I am seeing that will likely not be possible. I'm sure if I tell the organization I would like 2 weeks to provide the current company notice and I only provide 1 week and take off the other, that can look shady and I'm not interested in taking that risk for what could be a good role to grow from.

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Thoughts? I've never been in such a predicament lol

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