They CANNOT prohibit you from playing anyone (assuming the insurance is aligned). Now will those games count towards rankings? I can almost guarantee a no (at least from the ACHA).
Look at the DVCHC from last year. Salisbury operated two teams- one was ACHA D3, the other was non-ACHA yet the played in the same “league” (divisions different) as the non-ACHA team played a regular schedule vs ACHA teams with ZERO issues. The games did not count for rankings of course, but a regular schedule was played.
I think it comes down to do those six games impact your ACHA games requirement. If you can afford to add a few CHF games then who really cares as long as everything lines up.