Actually it doesn't strike me as particularly difficult at all. the pre-WWI UK did not yet have universal sufferage and even with the somewhat broad property qualifications that were applied (~40% of Englishmen could vote) almost no Africans or Indians would have been able to vote. The addition of a literacy test would have removed any threat that the Empire was about to be overrun by its colonials.
Of course, as attitudes changed the franchise would be opened to more of His (I had to correct my natural tendency to say Her) Majesty's subjects of all colors, but early 20th century imperialists need not recognize that initially.
It's definitely an interesting idea. You would have to largely remove the racial discrimination by a lot of the white settlers. I.e. while the black population could be disenfranchised for the moment continuing to treat them with such open contempt would be fatal. Coupled with making jobs and social status available for those who take up western education and demonstrate their loyalty to the empire and you could build up a substantial loyalist bloc. Coupled with the tribal divide affecting other elements full British citizenship could be seen as the best option not just for individuals but also for the population as a whole by many.
In the longer term you would have the problem of managing the adjustment of attitudes to accepting racial equality. Also the fact that most resources would still be owned by the relatively small minority unless there were major changes to reduce inequality.
However if you could manage to make one or two stable dominions in southern and eastern Africa you could add very valuable resources to the 'British' empire. The area could be very rich in terms of minerals, potential industrial wealth etc. Also they could quite possibly absorb areas such as the former Portuguese colonies and the Belgium Congo.
Steve
Steve