I'm currently seeking a job in a completely different field than I'm in now and I have some reservations about updating my LinkedIn profile to show my qualifications for the job I'm pursuing due to having current supervisors as connections. Should I update it anyway, or stick the resume route?

Update it. Most people don't actively monitor LinkedIn that closely. Also, you can turn off your send update settings in LinkedIn. It's pretty straight forward (right-hand column of LinkedIn) Move the "Notify your network?" option to "No".
We also cover this in a bit more detail on SE10. Jump to the relevant discussion on Soundcloud here https://soundcloud.com/twoguysoneshow/se10-so-much-to-discuss#t=1:05:23

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