I may be wrong, but I get the impression that you aren't dissatisfied with your current salary, hours, duties, stress levels, responsibilities, or job security. I'm sure it isn't perfect, but no job ever is. Anyone that tells you their job is perfect is trying to convince themselves. Instead, it sounds like you are only dissatisfied with the title and your ability to be promoted.
I've always viewed title changes as completely immaterial. Literally meaningless. Banks and larger corporations are notorious for making everyone a "Vice President in charge of _____." They do that so they can give a title that makes you feel important and increase job responsibilities without increasing your pay. If you look at may large banks, there are literally thousands of "Vice Presidents." I have a friend that worked for AT&T in the 80's. He was 24 and he was "Co-Vice President in charge of Finance Reporting in the Audit Department." All boiled down, he was Vice President of his job . . . which is nothing. The running joke at AT&T was that if you threatened to quit, they'd make you "Vice President of this Pencil" or "Vice President of this Typewriter" in an attempt to get you to stay. You'd be surprised how often it worked. To me, the only reason you need, or should seek, a title change is either: A) because it comes with more pay and better perks, or B) because you need to feed your ego. The second is somewhat pointless to me, but that's me.
Keep in mind that title changes and promotions alone often aren't all they are cracked up to be. They often come with increasing pressure, stress, and responsibilities. They also often come with more pay, but many people that take promotions later believe the increased pay doesn't justify the increased work and stress. If you're using it as a means to FIRE earlier, it may be worthwhile. But consider it.
As far as the new job goes, if you're only dissatisfied with the title and promotion possibilities, I don't think its worthwhile to change jobs. But even if you believe it is, you should be looking for a job that matches everything you have now PLUS gives you a little more. The new job should match your NET (not gross) pay after COL expenses, match your hours, stress level, ect. while giving you something more (either pay, title/responsibilities that lead to making you more marketable, security, long term growth, or something else). With a COL at 3x higher than where you're at now, that should come with a massive increase in salary in order for you to consider it.
That's my take though.
