The Logical Positivists did not accept synthetic a priori knowledge. They accepted only Hume's Fork, two kinds of knowledge, as you suggested in the question.
Logical Positivism was not a single shared opinion, but a variety of opinions and arguments under a shared general approach. We can take A.J.Ayer's Language, Truth And Logic (1936) as one representative. Ayer argues that all the cases that Kant took to be synthetic a priori are in fact either analytic or a posteriori. Ayer takes Kant to have been misled by the dominance of Euclidean Geometry and of Newtonian mechanics in his time. The discovery of Non Euclidean Geometry, the gradual refutation of Newtonian mechanics and its replacement, have taught us better. Every geometry is, within its own framework, simply a consequence of its axioms, and in that sense analytic. The fit of a geometry to the world, on the other hand, is a purely empirical, hence a posteriori matter. In any case, there is no place for the synthetic a priori any more. Ayer summarizes (p.80):
And thus we are able to dismiss Kant’s transcendental aesthetic without having to bring forward the epistemological difficulties which it is commonly said to involve. For the only argument which can be brought in favour of Kant’s theory is that it alone explains certain ‘facts’. And now we have found that the ‘facts’ which it purports to explain are not facts at all. For while it is true that we have a priori knowledge of necessary propositions, it is not true, as Kant supposed, that any of these necessary propositions are synthetic. They are without exception analytic propositions, or, in other words, tautologies.
