It's very difficult to do intarsia in the round, and you've figured out why, unless the main color is carried across as part of the motif. You can work back and forth and seam up one side and do whatever pattern you want.
If you carry the mc as part of the pattern, you can knit across using the mc and contrasting color. For the next round though, the cc yarn is in the wrong place. The mc yarn will have been knit around back to the beginning of the motif. What you would do then is to knit the mc stitches and slip the cc stitches. Turn your work and slip the mc stitches as you purl your cc stitches.This puts the cc stitches back in the right place for the next round.
After all the stitches are knit, you slip them all and continue to knit with the mc on the other side of the motif.
There is a way to knit back and forth yet connect the sides to be in the round, but I don't remember that technique off hand.