Duncan,
The usual BUDA format is, on the 36 hole days, to play one round of 4ball and one of 4somes. This gives everyone (especially the 18 holers) the chance to play the course and a quicker round as well for the 36 holers. I for one, and much as I enjoy foursomes, would like to play my own ball for 18 holes at a course I have never seen before and I suspect others will feel similarly.
And one of the best things about the best BUDAs is playing with a number of different partners/opponents. Anything that makes mixing the pairings up even a little harder should be avoided.