Depending on the type of analysis that needs to be done, I will warn against one team critiqueing another team’s work. No matter how well intentioned, this will only come off at best as of little consequence and at worst, vindictive. Especially if the critiqueing team’s own work was sub par. You do not want to create any animosity in a workshop. You won’t ever get hired by that company again. I saw it happen once. The critiqueing team’s idea didn’t make the cut, so they independently decided to rip on all of the other ideas. Ackward was a complete understatement.
The analysis should really be done by a neutral third party, you. This also gives the benefit to your company of showcasing your skills.
But if you are just plain lacking the resources, your best bet is to keep the ideas and analysis completely anonymous. The critiqueing teams do not know the authors of the ideas and the authors of the ideas do not know the critiqueing teams. But since these people work with each other, that may prove difficult to do.
Or as you said, have the teams self critique.