That stirrer will work fine, but it's probably overkill for this application. That is more for degassing. A simple long handled spoon would also work and be cheaper. The handles of spoons like this will fit through the neck of a carboy.

There is no other way that I'm aware of for a home winemaker to reliably check the completeness of MLF then doing a chromatography test. They are expensive, but usually come with enough supplies to run many tests. And I think the process is kind of fun. But I'm a chemistry geek.

This kit is a little cheaper at $60: hhttp://www.piwine.com/chromatography-test-kit-vertical-paper.html