Yes, the lab-frame muon free decay lifetime in a heavy muonic atom is probably extended by relativistic effects, but the muonic atom lifetime actually gets shorter for heavier nuclei.
The shorter muonic atom lifetime is because heavier nuclei have larger electric charges ($Z$) which make the orbital radius of the muon smaller so it spends more and more time inside the nucleus (which is also getting larger as $Z$ increases). Instead of decaying it becomes more likely to disappear by interacting with a nuclear proton via
$$\mu^- + p \rightarrow n + \nu_\mu$$
As shown in Figure 2 of "Study of nuclear properties with muonic atoms", the lifetime of muonic atoms rapidly decreases as $Z$ increases. This muon capture processe dominates for atoms with $Z>10$, and the lifetime of the heaviest muonic atoms (e.g. lead, uranium) is about 80 ns. The lifetime stops decreasing once the muon orbital radius is small enough that it spends essentially all its time within the nucleus. The radius of a uranium nucleus ($Z=92$) is about 6 fm, compared to the naive muonic uranium Bohr radius ($a_\mu = 260/Z\,\textrm{fm}$) of about 3 fm.
The free muon decay lifetime was measured back in the 1960s by looking for the energetic electrons from free muon decays, and this lifetime agreed with relativistic theoretical calculations. For example, the theoretically expected time dilation for lead was 16-20%, and the experimental value was 14±4%. Not a very precise test, but enough to strongly suggest the atomic muons are subject to time dilation.
