Due to Mazur, Akbulut and Kirby and many others, there are many examples of integer homology 3-spheres which bound contractible 4-manifolds given by attaching a single 2-handle to $S^1 \times D^3$ which algebraically intersects the one-handle once but maybe geometrically many times. Many of these can be distinguished from the 3-sphere by things like the Casson invariant and other invariants specifically developed for integral homology $S^3$'s for instance in https://arxiv.org/pdf/1508.01491.pdf
I would like to know how to distinguish the analogous constructions (which don't reduce to the above case) for homology $S^1 \times S^2$'s bounding homotopy $S^1 \times D^3$'s (homotopy equivalent not homeomorphic). Specifically are there any examples of knots in $S^1\times S^2 \# S^1 \times S^2$ which algebraically intersect both one handles once but not geometrically, that surger to (hopefully many or even infinitely for the application I have in mind) non-trivial examples of homology $S^1 \times S^2$'s? And if so, how does one distinguish these?
Note by construction these all bound homotopy $S^1 \times D^3$'s. It's very easy to draw such candidates knots, but they end up having a decent amount of crossings, and even then I don't know what tools I would use to distinguish their surgeries from $S^1 \times S^2$ or each other.