I’m relying on stuff in my head from decades ago here but:
The TM-192 gauss meter is capable of measuring electromagnetic fields of extra low frequencies (ELF) from 30Hz up to 2000Hz
That’s why the magnet creates a reading when you move it but not when static. Static is basically 0hz.
You can still use the meter though if you just need relative measurements rather than absolute. Possibly. If you can standardise how you move the magnet in speed etc (to give you a frequency), then you can measure the reading as you move it away (in distance) then move the magnet horizontally to generate a frequency.
The frequency will be low but the meter is measuring something, as you say, so if at 3cm, say, the reading is consistent, this approach could work.
If you want the actual Gauss measurement, it won’t work.
Edit: There are apps that might work.
https://play.google.com/store/apps/details?id=com.keuwl.gaussmeter