Is it improper to notify party manufacturing a knock-off of product with PATENT PENDING of POSSIBLE or POTENTIAL patent infringement while a patent is pending? What if application has been requested non-published? Product clearly states PATENT PENDING and there is an active application filed but not published. Is it required that the application is published to put copycat on notice?
You cannot take any action until your patent has been granted. By 'knock-off' do you mean the idea was stolen from you? Did they start manufacturing after your patent 'priority date'? If you applied for a provisional patent (and linked your patent application to provisional), did they start manufacturing after the priority date of this provisional patent? If yes, you can take legal action against them, once your patent is granted. If no, you should have listed the 'knock-off' as your prior art - and considering how similar the product is to your patent description - your patent may not be granted.
Can you notify them of the possible infringement of something that may become a patent?
Is it improper?
Maybe. But then, why would it be?
Is it a good idea?
You cannot exercise any rights against them (yet). So what happens if you notify them? Best case - they don't care. Worst case: They tell you that they have an older patent / ip-right and sue you for infringing their rights AND second worst case: they start submitting third party observations against yur patent which leads to a delay of the granting of your patent and might make the scope of your claims more narrow. Then when/if your patent is granted, they will opose it (=> post grant review/reexamination).
So what should you do in this situation?
One possibility is to try to get a utility model to have an enforceable right fast. The other possibility is to to push the USPTO (consider the patent prosecution highway) and try to get a patent fast.
Then you can contact them. (Wait 9 months, you don't want a reexamination/post grant review)
Anyways, as this patent (application) seems to be of some value, I'd suggest contacting a good patent attorney, they should tell you something similar and help you with your decision.
I don't have a complete answer to your question but until a patent has been granted, it is in legal limbo because you don't know which, if any, claims will be accepted in the end. Once granted though, if the company is still manufacturing it, you certainly have a case. I don't know if you could sue retroactively due to product manufactured during the pending phase.
You can do it yourself, I believe. Send it certified or have some form of proof. I have phone calls and emails.