I'd say nether of those. Conker was never a character Nintendo was fond of once Twelve Tales was retooled. Unlike Banjo, I don't believe Nintendo even had Conker's trademark. As for Battletoads, that series was absent during Rare's heyday with Nintendo. It's mostly known as a third party title for the NES/SNES, Genesis and the arcades. Of course that doesn't prevent his inclusion and admittedly, Rash would be pretty damn cool.
If I had to pick a second Rare rep after Banjo-Kazooie, I'd say Joanna Dark and Fulgore are the two that make the most sense. Both characters were heavily promoted by Nintendo back in the 90's. Fulgore was the face of KI and was all over Nintendo marketing material.
As for Joanna, she already had an item back in Melee. That, plus the Proximity Mine from Goldeneye debuting back in Smash 64 shown Nintendo had big interest in these series.
If I'm being honest though, I'd just take Mumbo Jumbo and Grunty over both. The more Banjo the better.