Well, here is what I would do...
1) find out if there are who you report to and his/her title. If his/her title is the one you used to have, you won't get it. Skip to #5 If not, keep reading.
2) If your potential boss does not have your old title, does anyone at that company have a similar level of title to your old one? What titles report to them? Who do they report to? If there are no others with a similar title to what you expect, it may mean that giving you the title would mean that they have to distribute new titles to everyone or run the risk of people leaving. Don't go there. Skip to #5 If not, keep reading.
3) If there are no other employees at the new company with the same title, AND you would be reporting to someone higher than to whom others with similar titles to your offer, then you could ask for the higher title and justify it by virtue of the higher reporting line and no other people holding the same title. See also #4
4) if there are others with the same type reporting lines as you, who hold the title you held at your old company but would not hold in the new company, by all means ask for the higher title. Can't hurt.
5) if #1 or #2 are applicable, but you really want to work for this company, take the job at the lower title. If you want the title more than the job, hold out. If it were me, and I was burning precious FIRE assets waiting for the right opportunity, had 5 years or less to go, then I would take the job and scr*w the title. I have had several acquaintances who have been maybe 5-7 years away from FIRE who lost their jobs, held out too long, and squandered their FIRE assets. One was out of work for 3.5 years, eventually took a lower paying job with a lower title because he was running out of liquidity and burning assets too fast, and has now lost his job again and has had to settle for yet a lower position with lower title and lower pay, just to survive and keep the kids in college. Don't let this happen to you. Now he won't be able to retire for at least 10 years, and by that time he will be eligible for social security, and will be about 66-67.
Oh, how do I know stuff about points 1-5? Well, as a senior manager, I have had to consider these kinds of requests before, and have had to turn away good candidates because they wanted titles that were not in accordance to our org structure...you know, when the candidate wants to be CFO and was CFO at his last company (that was 1/20th to 1/10th our size), and wanted a CFO title more than a higher paying job, even though his new boss was our current CFO. That said, I have been willing, on a couple of occasions, to raise the title if there were no conflicts created by doing so.
Hope this helps
Find Joy in the Journey...