My digital calliper is Site brand. I've just tried it against a new set of mandolin strings and can not an accurate enough reading with it - I'm usually under reading by 0.0001". The one is also prone to (I suppose) "slipping" - you need to keep checking it's zeroed.

I don't know if any would do the job you want but I'd suggest that is one to avoid.

I have checked the strings against a Sealey digital micrometer and they agree within a few ten thousandths. I think that one cost me around £50 so over your budget.

I would think you would be OK with a mechanical micrometer. Have a look at tooled up.