This doesn't really answer your question in a qualitative way but could be useful data(?) anyway.
For me, it goes the opposite way–it's easier to not eat meat right now when I tell myself I'm not giving it up forever, just until there is an ethically acceptable source.
Also, I expect that if an alternative to factory farming becomes commonplace, the next generation's view of factory farming will shift to something like "how could this ever have been accepted", and I kind of don't want to be on the wrong side of that.
These are two pretty idiosyncratic potential cultured meat-based reasons to not eat meat now. But on the other hand I'm someone for whom your interlocutor's argument doesn't have an impact, which might help you decide how prevalent people for whom it does have an impact are.
I think this would be an unusual way to think about the issue, and is likely to make very little overall difference to animal welfare.
Consider the population of Americans (I'm using my country as one example, with made-up statistics) who don't care how much meat they consume. Let's say that's 90% of the country.
Of the remaining 10%, what fraction eat less meat for religious reasons that won't be affected by this argument? What fraction eat less meat for health reasons? What fraction eat less meat because they don't like the idea of hurting animals, even if the overall "problem of animals being hurt" may be "solved" by cultured meat (according to their view of the world)?
None of those people should care about this argument. The only people who seem likely to eat more meat as a result of cell-based meat development are those who currently eat very little meat because they don't like the idea of contributing to the abstract "problem of factory farming", rather than caring about the suffering of particular animals, religious prohibitions, their own health, etc.
These people probably exist, but I'd think that they are very, very rare. (Probably overrepresented in EA, but still rare.)
I overall agree that the argument isn't enough to move the needle.
I'll just say that I think 90% is too high for people who don't care about about how much meat they consume. I think people's views on the issue are more complicated. I think there's a large group of people who have a general notion that eating meat is unfortunate, but don't reduce their consumption because it's not a thing for their ingroup, and also they bristle at the notion about someone else telling them what to do. Kind of similar to how lots of people think that their clothing is made unethically in sweat shops, but they buy it anyways.
If I had to choose a number of number of people who don't care how much meat they eat, it would be closer to 55%-60%.