In this answer, Gerald Edgar mentions that Haar measure is naturally defined on the $\sigma$-algebra of Baire sets (the smallest $\sigma$-algebra that contains all the compact $G_\delta$ sets) of a locally compact group and that the uniqueness of Haar measure can fail for larger $\sigma$-algebras. I wonder if there is a nice example of this.
Curious, I skimmed through Halmos's classic Measure Theory, and I found that he proves the existence and uniqueness of Haar measure for the slightly larger $\sigma$-algebra generated by all compact sets. (Confusingly, Halmos defines Borel sets to be those in this $\sigma$-algebra; I will stick with the usual definition of Borel sets.)
Is there a nice example of a locally compact group where the uniqueness of Haar measure fails for the $\sigma$-algebra of Borel sets — the $\sigma$-algebra generated by open sets?
To dispell some potential confusion (see comments by Keenan Kidwell and Gerald Edgar) Haar measures are not required to be regular (for the purpose of this question).