Look at the specs of your VHF FM Ham rig. It says bandwidth is
16K0F3. That's 16 kHz for 5 kHz of Dev. at 1 kHz modulation
frequency.

So to stay within 5 kHz bandwidth you need to reduce the audio
bandwidth and dev A LOT less than 5 kHz dev.

Consider 900 MHz and other 12.5 kHz channel radios that are 10K0F3
and sound like dirt.

There is another "gotcha" in your description above. If you vary
the
carrier at, say, 1000 Hz, from say, 146.999 to 147.001, the actual
bandwidth
will be somewhat wider than you expect, and it will be dependent on
the
frequency of the modulation. I know this doesn't make sense, it
has to do
with some weird math. If you studied Fourier series back in school
it was
some abstract mathematical thing that had nothing to do with the
real world.
Well, guess what. Fourier has everything to do with radio! The
result is
that to stay within the 5 kHz bandwidth, the highest modulating
frequency
has to be somewhat lower than 5 kHz. This is one reason why the
FCC
prohibits amateurs from broadcasting music; reasonable fidelity of
music
requires higher bandwidth than voice.

..