BlockBandedMatrix actually stores the column-blocks in a BLAS/LAPACK column major format. So it’s inversion uses standard LAPACK dense QR. (Gaussian elimination with pivoting is not used as it may cause fill-in.)
To support the symmetric case one would want to use a Symmetric(::BlockBandedMatrix). It’s not so clear how to use the storage format with LAPACK because the column blocks are rectangular (so not symmetric).
Note a basic block Cholesky is already implemented that supports block-bandedness: https://github.com/JuliaArrays/BlockArrays.jl/pull/126 But will need some work.