We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of their quotient admits a simple closed form. This allows us to prove several conjectures of McIntosh on asymptotic transformations of $q$-series. Moreover, our results extend some identities of Ramanujan and McIntosh.