Appropriate to mention using a lab's dataset when seeking grad student position in that lab?
Interest in and knowledge of a potential advisor's work is a big selling point for a prospective student, and this is a great way to do it.  I would definitely recommend playing around with the data, and it becomes a great way to connect with the professor, providing a basis for conversations while demonstrating your skills and interest in the work.  Even if what you can do is quite limited, your willingness to "get your hands dirty" and actually dive into the data is a good thing.

Do this as a learning experience, not to discover something.

You mentioned "I'm afraid of appearing disrespectful or full of hubris", and you are right to be concerned about this.  There is a big danger of doing so if you come to this professor claiming to have made a new discovery.  It could come across quite poorly, almost appearing you think you know more than the professor.  And unless you are an absolutely exceptional student, you probably aren't yet fully equipped to evaluate whether you have really discovered something interesting and valid.

Instead, I would to this as a learning experience, and start a conversation with the potential advisor framed accordingly.  Ask questions if there are things you don't understand.  Or say, "I did some analysis like this, is this valid?" And so on.

Even if you do think you have made an interesting finding, frame it in the same way: you have done some analysis as a learning experience, and want to discuss it with the professor to see if you are doing things the right way.  If there is a real discovery there, let them come to that conclusion.  At this stage in your career, it's much better to take a humble stance of desire to learn.

Also, this is a great chance to learn about this professor as a potential advisor. Are they interested in discussing what you are doing and answering your questions about their work?  Are they supportive of your interest in pursuing further study?  Do they make time to talk to you?  If so, there is a great chance that they would be a supportive advisor.

Conclusions:


Your interest in the professor's work, initiative to explore the data yourself, and any skills you demonstrate in the process all sell you really well as a potential PhD student.
This learning experience will be valuable for you: both in gaining necessary skills, assessing whether you want to do further research in the area, and whether you want to work for this advisor specifically.
Actually making new research findings from the data should not be your focus.  If you don't have the skills yet, acting like you think you can do this looks bad.  If you are the rare undergrad student who can operate at this level, let the prof discover it rather than claiming it.