This noun names the angry, sharp quality of an acrimonious talk or relationship — the bitterness in the air. It is rare and formal, even more so than acrimonious. Most writers say acrimony, which is the usual noun, or they keep the adjective. You might meet acrimoniousness in careful, academic, or legal writing that wants to name the 'being-like-that' as a quality, sometimes in a list next to other -ness words. Though actually, it does not add a new meaning. It is just acrimonious turned into a thing you can point at. If you already know acrimony and acrimonious, you already have what a learner needs; this form is a stiff cousin, not a different idea.

火藥味、尖銳的敵意（罕見的名詞說法；一般用積怨、火藥味就夠）