Is it appropriate for reviewer to offer cooperation on a reviewed paper?
Since this is your first review, you probably ought to keep the following in mind:
It is not the job of the referee to fix the paper, it is the job of the referee to provide grounds for the editor to decide whether to accept or reject it. Presumably you have research of your own, and while it is no doubt very kind of you to offer very detailed instructions for how to carry out a new analysis, this is ultimately not what you are tasked to do.
What you can do in your review (which is to be addressed to the editor), is to point out that their current analysis has a flaw, and whether or not you think the paper is publishable with this flaw. If not, then you can suggest that the authors (and not you!) revise the paper using a better method. Here you can give a comment stating your suggestion, and point them to a reference.
And to answer your question concretely: No, it is not appropriate. It would place you in a double role, where the authors might feel pressured into accepting you, to have their paper accepted. If you are very keen on publishing your method under your own name (provided that it is, in fact, novel), you have to wait until their paper is published, and then write a paper of your own. Note that you cannot use privileged information obtained as a referee.
Finally, don't sign your reviews. I would hope that the editor removed your signature. If you don't like giving blind reviews, you should abstain from accepting them, and provide reviews on open platforms which allows this. Like it or not, blind reviews is the name of the game.