By definition, Brønsted-Lowry acid-base reactions require that a proton (H+) be exchanged in the reaction thus forming a conjugate acid and a conjugate base:
HA + B --> A– + HB+
acid base conj. base conj. acid
Clearly, this is not the case here, so your reaction does not fall under the Brønsted-Lowry definition of A/B reaction.
By definition, a Lewis acid base reaction will require a base to provide electrons to an acid in order to form a bond between the two compounds, thus creating a new compound called a Lewis adduct:
A + :B -- > A–H
acid base adduct
In many ways, your reaction does look like a Lewis acid base reaction, where the acid is the barium ion (short on electrons), the base is the sulfate ion (electron-rich), and the adduct is the barium sulfate compound.
What looks "strange" to most students is the fact that most Lewis A/B reactions are presented as forming an adduct containing a coordination bond that it significantly covalent. However, the bonding in barium sulfate is also presented in most textbooks as being essentially ionic. This is however not entirely the case, as most bonding does not purely categorize as either ionic or covalent, but is, in reality, a mix of both.
When you look at examples for Lewis acids and bases past most textbooks, you will find that:
- complex (an)ions, such as the sulfate ion, are indeed considered Lewis bases
- alkaline earth metals, as well as hydrogen ions and alkali metals, are considered hard Lewis acids.
So, to sum it up, your analysis is logical. The reaction you have does represent a Lewis acid-base reaction, although not necessarily among the most typical ones encountered in textbooks.