We investigate the interplay between two fundamental mechanisms of microbial population dynamics and evolution called dormancy and horizontal gene transfer. The corresponding traits come in many guises and are ubiquitous in microbial communities, affecting their dynamics in important ways. Recently, they have each moved (separately) into the focus of stochastic individual-based modelling (Billiard et al. 2016, 2018; Champagnat, Méléard and Tran, 2021; Blath and Tóbiás 2020). Here, we examine their combined effects in a unified model. Indeed, we consider the (idealized) scenario of two sub-populations, respectively carrying 'trait 1' and 'trait 2', where trait 1 individuals are able to switch (under competitive pressure) into a dormant state, and trait 2 individuals are able to execute horizontal gene transfer, turning trait 1 individuals into trait 2 ones, at a rate depending on the frequency of individuals. In the large-population limit, we examine the fate of a single trait i individual (a 'mutant') arriving in a trait j resident population living in equilibrium, for $i,j=1,2,i \neq j$. We provide a complete analysis of the invasion dynamics in all cases where the resident population is individually fit and the initial behaviour of the mutant population is non-critical. We identify parameter regimes for the invasion and fixation of the new trait, stable coexistence of the two traits, and 'founder control' (where the initial resident always dominates, irrespective of its trait). The most striking result is that stable coexistence occurs in certain scenarios even if trait 2 (which benefits from transfer at the cost of trait 1) would be unfit when being merely on its own. In the case of founder control, the limiting dynamical system has an unstable coexistence equilibrium. In all cases, we observe the classical (up to 3) phases of invasion dynamics à la Champagnat (2006).