Even if there are minor negative short term effects (and while there almost certainly are >0 people in the world following the cited logic, I'm sure they're responsible for far less than even 0.1% of global meat consumption), it still seems to me like the most likely solution to factory farming in the long term, and thus the expected benefits of cultured meat very vastly outweigh the cost that is implied by that argument.
1) I believe most ethical vegetarians avoid meat in order to not actively cause any harm to animals, and not so much in order to solve factory farming. And for the former, the advent of cultured meat in the future doesn't make that much of a difference for their present behaviour.
2) People committed enough to actually think about how their actions contribute to creating a more vegetarian (or at least factory farm-less) world, and thus people who would in theory be affected by the given argument, probably aren't the same people that would think "oh well this issue is being dealt with by others already, nothing to do here". Plus 1) still applies here, as people with such a level of commitment almost certainly also want to avoid personally causing harm to animals.
3) the exception to 1) and 2) may be a few effective altruists (or people with similar mindsets) here and there, who get to the conclusion that sticking to a vegetarian/vegan diet is not worth it for them personally given the apparent tractability and non-neglectedness of the problem, but we're probably talking about dozens or at most hundreds of people around the globe at this point, and if they actually exist this would even be a good sign, as the reason these people would make this decision would be the fact that, well, cultured meat solves the issue of factory farming in such an effective manner that their personal contribution via ethical consumption would have a smaller marginal impact than whatever else they decide to do.
Admittedly a lot of speculation on my part, but what it comes down to is that the argument, while probably playing some non-zero role, just hasn't enough weight to it to justify changing one's view on cultured meat.
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.