Is there a way to prevent the production team from messing up my paper?

Is there a way to prevent the production team from messing up my paper?

Production teams don't want to mess up papers; the ones that do are performing poorly. Unfortunately, poor performance seemingly occurs amongst all publishers, regardless of prestige. So, you cannot prevent the production team messing up your paper.

Requesting that the production team, do not edit the text, makes the team's job impossible. Attempting to pre-empt mistakes, e.g., by instructing that they do not remove the separating lines from the tables, as they convey important information, won't help much, since you cannot pre-empt all mistakes. Moreover, such contact with the production team will suggest that you have a low opinion of them, which isn't in your favour.

Following the publisher's prescribed style may reduce errors, since the production team won't need to make as many edits.

All is not lost, since mistakes can be fixed before publication, albeit this is expensive (in terms of time). In addition, preprints or technical reports (which aren't edited by the production team), can be made public.

Generally, publishers will send you a version with the changes introduced by the copyeditors, and ask for your approval. Did they omit this step? If so, if it were me, then personally I would be very upset, enough so to complain to the editor-in-chief. Although standards differ in different disciplines, I concur with Bryan Krause's comment that this was extremely rude of them.

If it is not too late (i.e., if the paper has not yet been physically published), then I would instead recommend that you contact the journal and request that they start over. I would also request a detailed list (or annotated version of the paper) explaining every change that was introduced.

In general, you can't reasonably ask copyeditors to not edit the text of your paper. But you can certainly ask that they send you a copy of the edited paper, complete with a detailed list of changes, and that they await your corrections or approval before proceeding.