Not cheaper in this case, but I found them more approachable/flexible
plus the council's planning dept had already showed what a mixture of
helpful vs stubborn behaviour they were capable of (depending who I saw).

That was the case for us, their fees were dearer than the LBC but the saving of several hundred pounds came with the actual garage. The garage we ordered was a standard design which was 2m longer and with some difference in window positions and beyond the BC free allowance of 30sq mts. The local BC wanted the engineering drawings for garage with all stress calculations which the garage manufacturers were willing to supply at a price. The private inspector was happy to work from the standard design drawings which were supplied free and extrapolate them to take into account the differences. If I recall correctly this saved me £200+.

Richard