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Since the discovery of cosmic rays (CRs) over a century ago, their origin remains an open question. Galactic CRs with energy up to the knee ($10^{15}$ eV) are considered to originate from supernova remnants, but this scenario has recently been questioned due to lack of TeV $γ$-ray counterparts in many cases. Extragalactic CRs on the other hand, are thought to be associated with accelerated particles in the relativistic jets launched by supermassive accreting black holes at the center of galaxies. Scaled down versions of such jets have been detected in X-ray binaries hosting a stellar black hole (BHXBs). In this work, we investigate the possibility that the smaller-scale jets in transient outbursts of low-mass BHXBs could be sources of Galactic CRs. To better test this scenario, we model the entire electromagnetic spectrum of such sources focusing on the potential TeV regime, using the `canonical' low-mass BHXB GX 339-4 as a benchmark. Taking into account both the leptonic radiative processes and the $γ$-rays produced via neutral pion decay from inelastic hadronic interactions, we predict the GeV and TeV $γ$-ray spectrum of GX 339-4 using lower-frequency emission as constraints. Based on this test-case of GX 339-4 we investigate whether other, nearby low-mass BHXBs could be detected by the next-generation very-high-energy $γ$-ray facility the Cherenkov Telescope Array, which would establish them as additional and numerous potential sources of CRs in the Galaxy.