You could do this using skip patterns within the same Lighthouse Studio survey. But, we put a limitation in the software that you cannot skip directly to a conjoint question. The workaround is to place a text-only question as introduction instructions just prior to the CBC question. You can then skip directly to that question.
A second part of your question is how to make sure, say, exactly 1 of the three exercises is completed. You can do that by using the RandNum(seed, min#, max#) to draw an integer from 1 to 3. (See the documentation for more details).
Then, place pre-skip logic on the text-only transition screens that are right before each CBC section. If the integer doesn't match the value needed to flag that the respondent should take that CBC exercise, pre-skip to the intro question just prior to the next exercise, etc. The skips bounce the respondent to all three exercises, but only one pre-skip logic will come back true, indicating that the respondent should continue to the next question and complete that CBC exercise.