Even more confusing is adding "building" to the end as a museum is already a building.

Not quite. In English, a museum is usually understood to be a building, but it can also be a more abstract organization. In Klingon, it seems that {'angweD} has the abstract meaning as primary, and adding {qach} is the normal way to get across what in English is the default.

A museum can be any collection of artifacts or objects on display—even non-solid ones. The IT department I used to work for once had a museum: a few shelves on one wall with a collection of old computers, each with a placard describing it. It was an 'angweD without an 'angweD qach.

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SuStel
http://trimboli.name