Exactly. If this is possible then it is OK. If this is not possible then it's going to be a bloody pain.
The British Empire is complicated enough that it would need a DLC to do it justice. You have everything from informal empire over nominally independent states (e.g., Iraq), to democratic self-governing Dominions (Canada, Australia, New Zealand, South Africa), to colonies (Nigeria), to protectorates (Egypt).
You cannot simply handle Canada as a puppet because it was not a puppet, but handling Canada simply as an independent state allied with the UK misses out the depth of the co-operation between the two. Handling Egypt as British territory misses out the very real fact that Egypt's rulers had 100,000+ strong army and were to a degree sympathetic to the Axis.