Your guess is right! Both states are indirectly named after the Kaw people, with Arkansas using a French pronunciation. This pronunciation had been anglicized as "Arkansaw" prior to statehood before the current spelling became standard.

The main reason the pronunciation diverged from Kansas was that the Arkansas government legislated the current pronunciation in 1881. This was partly due to different pronunciation by the two senators.

​

Whereas, confusion of practice has arisen in the pronunciation of the name of our state and it is deemed important that the true pronunciation should be determined for use in oral official proceedings.

And, whereas, the matter has been thoroughly investigated by the State Historical Society and the Eclectic Society of Little Rock, which have agreed upon the correct pronunciation as derived from history, and the early usage of the American immigrants.

Be it therefore resolved by both houses of the General Assembly, that the only true pronunciation of the name of the state, in the opinion of this body, is that received by the French from the native Indians and committed to writing in the French word representing the sound. It should be pronounced in three (3) syllables, with the final "s" silent, the "a" in each syllable with the Italian sound, and the accent on the first and last syllables. The pronunciation with the accent on the second syllable with the sound of "a" in "man" and the sounding of the terminal "s" is an innovation to be discouraged.

Edit:I had a tour guide in Little Rock tell me the different spellings and pronunciations were used to indicate feelings towards slavery, similar to how Derry/Londonderry indicates political beliefs today. I haven't found any support for that specifically, but controversy over the name has been used to make political and social distinctions.

https://encyclopediaofarkansas.net/entries/arkansans-versus-arkansawyers-6438/