I think that we will find out more to this story as we discuss the possible "reasons" that Verizon would be unable to add the watch's IMEI to their system. There are really 3 possible reasons.
Reason 1: The watch and the associated IMEI was reported lost or stolen and it has been blacklisted. Reason 2: An Insurance claim (carrier) was made for this watch and the associated IMEI and it has been blacklisted. Reason 3: watch was originally purchased on an installment plan and the original purchaser stopped paying and it has been blacklisted.
@renzska, if the cause of your problem is not listed above than Apple and Verizon should be working together to fix this issue for you in 2-3 days. Worst case since your watch should still have a valid Apple warranty, Apple should cover this issue as a warranty claim and simply replace the watch. Do you have an Apple store nearby? If not you can always call Apple again and escalate your case.
The information you posted about Verizon claiming that you need to get a "Verizon" model is as you suspected just bad information from an uninformed CSR. There are only 3 versions of the cellular model available worldwide. There is only 1 version for use here in the USA by all 4 of the Major carriers: AT&T, Verizon, T-Mobile and Sprint. We have had reports from all 4 carriers where a rep is insisting that you must buy your watch directly from the carrier. They really can't explain why Apple would also sell the Cellular watch if it "must" be purchased carrier specific.
Now let's look at the issue being simply a poorly trained carrier CSR. What the rep needs to do is to "park" the IMEI and the EID numbers from the watch on to a new $10 Wearable plan on your existing Verizon billing account. Both the IMEI and the long EID numbes on located on the bottom of the watch's box. Once this is done you should be able to complete the rest of the setup using the Watch app on your paired iPhone.
A couple of things to also be aware of is that your plan must be a fairly recent plan and be a post-paid account and there are limits to the total number of phones and connected devices per plan. You must login to the MyVerizon mobile app as the Account Owner or Account Manager to add the NumberShare feature that allows your cellular watch to use the same phone number as your paired iPhone.
Dave