I don't see any reason you couldn't do this with one questionnaire if you want to.
To split the respondents, you'll likely want to use a quota question. You could give the quota two cells, both set to always qualify. Then if you randomize the quota cell order and set the quota to use least fill, you should end up with a fairly 50:50 split.
After that, it's just a matter of adding skips to your questionnaire. On whatever question comes before the CBCs, you can add skips that skip some respondents to the first CBC and some to the second, using logic like "QuotaQ = 1" and "QuotaQ = 2" (where "QuotaQ" is the name of the quota question, and the 1 / 2 are the cell values in the quota question). Then, at the end of the CBCs, you can add skips to bring all respondents back to the follow-up questions.
Note that Lighthouse Studio doesn't allow you to actually skip directly to or from a CBC question. If you add a text question to the same page, you should be able to use that for your skips instead of the CBC questions themselves.