How to center font with specified height

tngjstn,

As I (mis)understand it, the old cheat is to:

1) Tick Use Preview/Artwork Bounds in Edit>Preferences>General;
2) Select the live Type with the (normal) Selection Tool and Effect>Path>Outline Object;

3) Set the position to 0.

Obviously, you will need to untick Use Preview/Artwork Bounds for other purposes, but you can just tick that again when you need to edit (some of) the Type if needed to keep it in place; that may not be necessary because the changes will (mostly, and only if no letters exceed the hitherto vertical bounds) be in the horizontal direction.

1) - 2) will give you a Bounding Box that fits tightly round the actual set of characters in your live Type, and will ensure easy and accurate positioning, such as centring (both directions), and keeps doing so when you edit, without resorting to working (indirectly) with the Glyph (bounds) concept,

https://helpx.adobe.com/illustrator/using/snap-to-glyph.html

If you need it as non outlined text in PDF you will have to remove the effect; you can:

4) With the live Type selected, in the Appearance palette click the Reduce to Basic Appearance button.

What do you mean by "set to 0"? Where are you going to set that value?

You may want to clarify a bit.