The database Eighteenth Century Collections Online (ECCO) includes 34 editions and reprints of Lowth’s grammar. You can find these by means of an Advanced Search string that includes Lowth = author AND grammar = title NOT Ash = author. If you reorder the results according to Publication Date Ascending, the resulting order suggests that the second edition (1763) was published in Dublin.
This edition, as I explain in The Bishop’s Grammar (Tieken-Boon van Ostade, 2011, OUP), was not the second edition as such but a pirated edition of the proper second edition, published that same year by Lowth’s regular publishers, Robert and James Dodsley and Andrew Millar. This is an inevitable disadvantage of having to work with (otherwise fantastic) electronic databases, but it is important to be aware if this as misunderstandings may easily arise (and have done so) as a result.
As in the case of the ECCO editions, these editions are fully searchable. The Basil edition, published by J.J. Tourneisen, is not included in ECCO.