No, as per this page The non publication request, if filed, must be filed with the filing of the patent application. If the non publication request is not filed with the patent application, then the patent application will publish in due course. If you wish to maintain the secrecy of your patent application until it matures into a patent, the only other ...
If the invention is not disclosed in another patent application in another country or patent organization (EPO) that requires publication, then a request for nonpublication can be made AT THE TIME OF FILING. The application will then not be published until such time as it may issue.
37 CFR 1.213 Nonpublication request.
(a) If the invention disclosed in ...
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 ...
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?
Not published means not available for the public to see at all. The AIA 3rd party submission process window opens at publication and closes six months later or at first action on the merits. For an application with a non-publication request that window would seem to never open.