In other words, I want the Stroke to widen inwards and the outside edges of the object to remain unaffected.

Unfortunately, that's not possible yet. It's a much-requested new feature though, that's for sure! For some reason I'm thinking it's going to be available in 0.49, but I can't remember where I've read that. And I might have it confused with some other feature. But it's definitely "on the drawingboard"

I don't know exactly what your situation is. But there are 2 kinds of bounding boxes in Inkscape (the dotted lines which indicate the selection). It can be either a Visual bounding box, which is the default, and includes the stroke width in the dimension measurements; or it can be a Geometric bounding box, which is only the bare path, without stroke added into measurement. It's in Inkscape Preferences > Tools. There are some situations where it's useful to switch to geometric bb.

Without knowing more about what you're doing, I don't know if it would be useful in this case. But I just thought I'd mention it

There is a work around however.
1. Apply the stroke that gives the outside edge you want.
2. Duplicate your object.
3. Apply a stroke to the duplicate that is twice what you want the "internal" extension of the stroke width to be.
4. Select the original object and bring it to the top, aligned directly above the duplicate with the thick stroke.
5. Select the original and the duplicate with the thick stroke and do Object>Clip>Set.

A bit of a kludge, but it should do what you want.