What's the term for the inner perimeter of something?

This question presupposes a false distinction.

If the perimeter was a line drawn on a flat surface (cricket pitch, for example), a pedant might say you could only walk round it if every footstep fell directly on the line.

If the perimeter was, say, a line of close-set spikes sticking three inches out of the ground so you couldn't actually walk exactly on them, "walking the perimeter" would apply equally to just inside or just outside the line (or indeed randomly crossing over the line sometimes, thus including both).

It just so happens you can't normally walk the inner perimeter of a building, because there are usually dividing walls in the way. So by default it's assumed you mean the outer perimeter. If it is physically possible, and you want to say you're going to walk the inner perimeter, just say that. There is no more succinct way to express the difference.

If you are speaking humorously, you might say, "Now I'm going to walk the battlements."

Better than that I cannot do, because I know a perimeter is, geometrically, a line of zero width, on which nobody can walk, so no sensible person (whether a pedant or not) would ever talk about walking a perimeter. Perhaps they would talk of perimeter patrols.

Perhaps perimeter is not the word that should be in question. Instead, there may be a better word for walk.

For example,

"I'm going to pace the perimeter of the building."

If now you're walking inside and want to succinctly say that you are making your way through the building, visiting certain rooms and ending up in the same place as you started, you could say:

I'm going to make my rounds through the building.

The idiomatic phrase make rounds calls to mind a doctor going through the hospital and visiting each patient. The definition for the phrase says it means

to visit or call certain people or places

So if your trek through the building involves specific stops, this phrase would capture that meaning.

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Another option is that you are taking a tour of the building. This term does not only imply that one is a tourist in a vacation locale -- the definition includes:

a journey for business, pleasure, or education often involving a series of stops and ending at the starting point; also : something resembling such a tour

So you could say,

I'm going to tour the building.

I don't think you can do better than inner perimeter. If you have a sense of humor, you can use E-ring -- the Pentagon's inner perimeter ring.