Minis did race and rally with Webers. BMC Special Tuning sold Weber kits for Minis.
The Australian part number for a Weber kit for a Cooper S is C/HYL7007.
Lots of classes for racing and rallying didn't require homologation of parts. When the Minis were running in the Monte in Group 1 they had to run stock HS2 SU carbies. These were the homolgated Group 1 carby.
When they run the Monte in Group 2 and Group 5 they could run H4 carbies. You didn't need to run a Homolgated carby you could just run whatever fitted the stock manifold as long as it was from the same manufacturer. Later on they fitted split Webers to the stock manifold. (except they were not really Webers, they were BMC experimental carburettors that looked like Webers because otherwise they would have been cheating.

(They really just filed the Weber names off them.

))
Just because BMC sold Weber kits does not mean that they were ever Homologated for Group 1.
Looking at one copy of the Homologation papers (out of Peter Brownings book "The Works Minis") for the Cooper S shows the Homolgated carburettors to be the HS2. It shows 8 port heads and alloy 8 port heads for Group 2. There is no mention of carbies for Group 2 because they would have been free under the rules (or almost free!).