This myth is repeated every year. I would like to know the true source for this “minhag”. Reb Mendel of Rimanov never, ever said any such thing relevant to Tuesday of the week in which we read Parshas Beshalach. He was quite busy with Parshas Hamon, and strongly advocated saying it daily. This was verified with descendants of Reb Mendel who themselves come from Rimanov. I also heard from a quite prominent talmid chochom who claims that prior to a particular event (post-Holocaust), this minhag was never known, and that he was present when someone suggested he would dare to create this minhag. It is legendary that minhagim in which the segula is parnosoh tend to spread rapidly, or in the current vernacular, “go viral”. There is a reputable source from sifrei kaboloh about reciting it “shnayim mikro ve’echod targum”, but again no reference at all to any particular days. There is no explanation I ever heard (having researched the subject) for why Tuesday of Beshalach.
There is certainly nothing wrong with reciting it, but I would be very careful to avoid spreading a lie, that Tuesday of Beshalach is a segulah from Reb Mendel of Rimanov.
If someone can find a reputable source, obviously better than “I saw it on the internet”, I will be overjoyed to see it. Meanwhile, I have addressed this with quite a few major talmidei chachomim. None can verify that there is a source other than something that circulates among “hamohn am”.