Cant see an issue with your suggestion but this is the public sector you’re dealing with so be prepared for it to be dismissed out-of-hand.
I’m GPS tracked at work the whole time (courier) and it’s definitely helped catch out the workshy! So easy to catch out the time wasters who park up and have an extra break rather than get the job done (paid by the hour so longer day = more money).
There’s no issues with being tracked, you’re at work so they should know what you’re doing anyway. No different to remote monitoring of a workstation for example. The ones who will kick up a fuss are the slackers! Doesn’t have o be live tracking all the time either, so long as the data can be downloaded at the end of the shift the management can look at it. We had a few lads who thought turning off the data on our PDA’s would cover them, two got done for being at a football match 😆