I'm not sure if "the" lab is needed, or if it's simply that the top end nic handlers all perform a reasonable analysis of the nic content, whether done in-house or through an outside group (and I wouldn't care if it was from one specific lab or any lab with the right tools).
Using TPA for GC/MS (I hadn't known TPA did this until I saw the BE report, but, yes, it's listed under their available services

) is not bad, nor is having it done by someone else. If the focus is on just checking for the nic concentration, is GC/MS even a requirement, or would a titration performed by a professional be reasonable?
If someone ran a high-end test earlier, super, but I would absolutely expect that each bottle coming out the door can directly trace to a test result showing that the bottle has the labeled value. Any volume vendor would hopefully be working off of larger source containers, for example a 5 gallon mix of 60 mg/ml, which is used to create bottles of 60mg/ml for sale. The 5 gallon final mix container is where the lot # should come into play for outside use, the larger container of 60mg/ml from which bottles for sale are produced, tied to a single final test result. The label on the bottle should clearly have a lot # indicated, and it should be possible to look up the tested value. (There is a customer training aspect to this, people will need to learn that the 100mg/ml label may really test as 102, or 98, and still be within reasonable manufacturing tolerances -- surprise!)
Issues of cleanliness, material handling, etc. are separable.
While a vendor group checking to see that vendors are indeed doing the tests that they record, and is able to do an audit of a sample, is good, I personally wouldn't expect or require such things to occur all that frequently. The sealed bottles, chain of custody, sending things to a central authority to then pass along for formal tests is a bit more onerous or administration heavy than I would expect to be needed for normal operations, the kind of thing I would expect if a vendor has seemed to lapse than how ongoing activities would occur. My starting point is the idea that vendors are generally good people trying to do the right thing, but a bit of framework providing best practices would help.
(I absolutely agree with Jan that if a second tier vendor then does anything, in any way, to the incoming nic from the first vendor, other than strictly rebottling, then there should be a new lot number, with backup test, showing that the output is what it is said to be. The simplest example would be if a lower tier vendor buys quantity 100mg/ml, then does their own internal cutting to create a 50 mg/ml for resale -- that 50 should be checked, and have its own new lot number.)