Although Rare was entirely capable of transferring the data via controller pak, I find it unlikely they did. Just some quick thoughts about it though.
First, we are talking about Rare, and transferring data bewteen games without the controller pak would be the milestone. Fully utilizing the capabilities of the game console is what we've come to expect from them, DKC for the SNES being a perfect example.
Second, once the concept was proven to work, it should always work. There would be no need to code a backup data transfer mechanism into BK since the RDRAM holdover method already worked. Consoles aren't supposed to change, that's the benefit of developing for them.
Third, I think Rare would have needed to indicate Banjo Kazooie did indeed use the controler pak. The only indication is for the rumble pak, though.
BK Box Art
In comparison, Diddy Kong Racing was released in 1997, where the controller pak use is indicated.
DK Racing Box Art
However, the controler pak issue was discussed here in the forums several years ago. If I recall correctly, someone using a N64 emulator to play BK claimed the memory pak was written to. A .pk3 file or something with a similar extension was generated after a series of events. I don't think that ever went anywhere, and could very likely have been false information.
---