No, "double NAT" should not cause any issues. Device should be able to connect to RMS as long as it has internet connectivity to public internet (i.e. not located in "private APN network") and can reach AWS (Amazon Web Services).
Try to navigate to device's "System -> Administration -> Diagnostics" menu and initiate ping to "rms.teltonika.lt", when your router is using Mobile WAN. If pings would pass, then this might indeed be "port blocking" issue. If, however, pings would not reach "rms.teltinika.lt", then either whole connection to AWS is blocked, or your SIM card is located in private APN network. In both cases you could contact your SIM card provider with discovered information and ask why connection to "rms.teltonika.lt" is being restricted.
Communication with RMS happens via the following TCP ports: 80, 443, 15009, 15010