If a reviewer makes significant contributions to improving a paper, may he/she suggest becoming a coauthor?

At this point, I wonder what the best way to proceed is. Should I just submit the report with these suggestions?

This is indeed the high road to take here. As a reviewer, you explain your ideas on how the paper can be improved. The authors will or will not take all or some of them into account.

Asking for co-authorship in return for your service would likely be perceived as very cocky and unprofessional. I doubt that either the editor or the authors would be pleased about this or seriously consider taking you up for your offer.

I have been on the receiving end of such an extremely helpful review, and I am grateful for that. Since (s)he is anonymous all I could do was thank the anonymous reviewer at the usual "thank you footnote". That won't help her or him much, but if he or she reads it he or she will know that the help was appreciated.

However, the editor does know that you have written an exceptionally helpful review. Leaving good impressions like that does tend to come back at some point: maybe (s)he is in a hiring committee for a position you are applying for, maybe the journal is looking for new (associate) editors, ...

In line with the other answers, I agree that you should not suggest becoming a coauthor.

However, an alternate route could be to relinquish your anonymity by signing your review. This would allow the authors, should they choose, to acknowledge you by name or contact you directly.

There is the possibility that your choice to do so could be regarded as self-serving in this case, but there is clear precedent for signing peer reviews.