Would a suitably placed inductor stop audio feedback whine?
I was thinking of possibly using an inductor that coupled with the
impedance of either the mic or speaker, depending on inductor
placement, would have an L/Z time constant such that the cutoff
frequency would be approximately 1.5khz

There is no simple one-size-fits-all solution to this because there can be many, many wavelengths of signal (in the air) between the speaker output and the path audio takes to get back to the microphone. Just moving the speaker or microphone 1 cm can be enough to stop acoustic feedback then, a moment later, due to the audio being reflected back to the microphone along a different path, the signal coming back will sustain oscillation i.e. it becomes positive feedback.
In other words, a simple filter will not do the job effectively at all. It's not like an electronic circuit where negative feedback is applied; it's much more complex than that.
My first recommendation is to incorporate a voice operated switch so that when you are speaking, the amplifier is activated and when you stop speaking closes down thus preventing sustained feedback. This of course relies on normal speech levels being the dominant amplitude.
The next trick is a pitch shifter. I remember electronics magazines way back when publishing a circuit that shifted the whole vocal range by 5 Hz. This is a good way of ensuring that you can't get a sustained feedback but may generate other artefacts. I heard at the time that you could get another 10 dB gain from the amplifier.
I do recall that some megaphones used pitch shift to avoid feedback and I believe that some PA systems do the same.
Along the same lines as pitch-shifting is using a &quot;Dalek effect&quot; &quot;translator&quot; where you are removing a significant amount of natural spectral content of the voice and replacing it with something else. That &quot;something else&quot; can be an assortment of band-limited noise, regular oscillator tones or the output of a musical instrument (aka a vocoder). This type of circuit is generally referred to as a ring-modulator (that's what is used for Dalek voices). I've seen videos of the guy who plays Davros practising lines sat at a table with other Dr, Who actors and wearing the device with no apparent feedback problems.
There are also DSP chips that can eradicate sustained oscillations by modifying in real time the audio filtering algorithm to reduce gain at where it detects the sustained tone.
There is this paper that desribes how frequency-modulation is used to reduce feedback in hearing aids. It considers: -

Gain reduction techniques
Frequency shifting
Adaptive frequency compensation
Frequency modulation

Elliot Sound Products aka ESP have a very good article on the link called
Acoustic Feedback In PA Systems (With Special Reference To Frequency Shifting)

The article also goes on to show a circuit diagram of a frequency shifter: -

This circuit and another are on the project 204 page.