What the NVC provided in the past was the phone call-in to request your case number (which is different from the USCIS receipt number) and the invoice number (which is like the password for the case number to input to the CEAC tracker) after it was assigned by the NVC, which allows you to use the CEAC tracker for the status update of the case.
https://ceac.state.gov/CEACStatTracker/Status.aspx
And NVC may even mail or email (to the email address provided in the I-129F) the case number notifying that the case is transferred to the consulate. The case number is then used for the DS-160 and UStravelDocs.
In the absence of the NVC providing the case number, the Packet 3 notification would have the case number from the consulate sent theoretically to the K1 beneficiary (either with the email from I-129F or the beneficiary mailing address).
Keep in mind that after the NVC stage, theoretically all the steps remaining in the K1 process at the consulate stage are to be completed by the beneficiary (medical exam, MRV fee, interview, etc.), and the only task that the petitioner needs to complete is to send the beneficiary the I-134 affidavit of support, which the beneficiary takes to the interview. All these steps may be specific to the embassy/consulate. So for example, the steps in Packet 3 from Manila Packet 3 may be different from Warsaw Packet 3, etc. So follow your consulate site instructions--don't assume all the consulates' processing are the same.
As it seems with NVC no longer providing call-in support for the case number, then the only option is to wait for the Packet 3 from the consulate. If the wait time seems abnormal, I would call the consulate to see if they have received the case (using the USCIS receipt number) and see if the consulate could provide the case number and the Packet 3 notification? With the shortage of workers either at the NVC or the consulates, I would not be surprised if some cases slip through the cracks and the beneficiary/petitioner never gets notified? So I would stay proactive and contact the consulate if needed. But at the same time, I anticipate the K1 process to evolve and these steps may be very different in the future rendering any advice from old K1 applicants to be obsolete.
Invoice ID is not necessary for K1, as there is no need for K1 to ever log in to the CEAC portal to upload documents. That is only for immigrant visa cases like CR1. Only NVC case number is required to check K1 visa case status using the online tracker -- https://ceac.state.gov/CEACStatTracker/Status.aspx?App=IV
Note that though K1 is a non-immigrant visa, it is processed as "immigrant visa" until after the visa interview. So select "Immigrant Visa (IV)" as visa type on the tracker.
