OK, the "no need to test" line is one that a lot of us have heard from our vets. Some vets seem to think people won't want to bother checking their cat's BG levels often, and others seem to be convinced that it's safe to give insulin without knowing the BG level in advance. I'm not sure you're going to find many people here, if any, who would agree that that's safe for any cat. But you wouldn't give insulin to a human diabetic without knowing their BG level first, so I see no reason why it should be safe to do that to a cat either - even a fairly young child can tell you they're "not feeling good" if they're heading towards a hypo. A cat can't do that, so if anything I'd have said that knowing a cat's BG level is even more important.
It's possible that the shots she had yesterday might have contributed to her feeling icky. My vet told me they prefer not to give annual shots to a newly diagnosed diabetic, though I think that's more their opinion than anything I can find that says it's not a good idea. And with Rosa being an indoor kitty, they might have thought it wasn't as important for her to get her boosters as it would be if she was outdoors a lot. In the same way that some humans can have a slight reaction to shots, I'd have thought it's possible that some cats might feel just a little less than 100% in the day or so after they have their boosters.
The good thing with Lantus is that Chelsea doesn't need a lot of food on board at shot time. With the older insulins, it was important that the cat ate at shot time because the insulin would kick in fast and fairly hard and could drop them a long way quickly. Lantus takes longer to onset and is more gentle on their system, so it's not important for her to eat a lot at shot time. If she's not feeling great, maybe leaving a little food out for her all day might help...that way she can just pick at a little food when she feels like she can. You do, however, need to make sure that she's eating something at some point - is there a food that she particularly likes (even if it's not low carb for now, any food is better than no food) that you might be able to tempt her with? Even some canned tuna or maybe some cheese sprinkled on top of her regular food...anything you can find that will tempt her to eat for now is fine. Did you change her food quickly to a vet recommended one or anything? That can sometimes upset their stomach and cause vomiting.
And for now, as long as you have time to just sit with her, maybe that's just what she needs today. If she can just relax with you then maybe she'll feel a bit more like eating. It's unlikely that she has ketones given that her numbers don't sound as though they've been high, but if you can check for them just to make sure there's not something else going on, that would be a good idea.
Even longer term, you can free feed with wet food if it turns out that she doesn't like to eat a big meal all at once - lots of small meals are less strain on her pancreas. And I've never had any success insisting the cats here eat when I want them to - they all free feed all day.