I'd like to essentially use the "put on path" function, but have the text rest precisely through the center of the path rather than sitting on it. (or know of a way to accurately create this)

I'm a bit of a stickler for precision. I know there are ways of just eyeballing this to look good enough but it would comfort me to know that it was precisely in the center (ideally being able to type the exact digits in, e.g., if I want a perfect circle I could type in w:100 h:100 and know it was correct rather than eyeballing and being 1 or 2 out).

The first screenshot is from GIMP which seems to do this by default and shows what I mean. I want it through the path, not on it. Screenshot 2 is what Inkscape does of course.

I have attempted to search for something like this but haven't found anyone talking about it. All you get is people showing how to put it onto the path rather than through it.

Thanks for taking the time to consider my question and for any insight offered.

I don't know why this works. 

If your document is set up in mm. 

Write down the height of you text in mm.

Text > Put on path.

Then double click the text, Ctrl + A to select all, and set the vertical kerning value to half the text height with no units.

Thank you for your help, inklinea.

This worked.

My option:

write the text of a certain size (es. 10mm height)

draw the circle, say 100x100mm

duplicate it, put the text on this

resize this reference circle to 95x95mm. (re-center it if necessary by snapping together the 2 circle centers).
Convert the text to path

delete the reference circle. (at this point you'd have the text halved in 2 by the very first circle).
hope this is what you're lookin' for.