Hmmm... in the standard player arsenal, Transports, Tachyon Drones, Decloakers, Neinzul Enclave Starships and Scout Starships are immune to mines. Have you checked whether suiciding a Tachyon Drone is worthwhile? They're under Decloakers on the build menu, IIRC, are much cheaper than a scout starship, and can be built on demand (in supply) just outside the wormhole. In fact, it seems like they were meant to be the original wine sweepers.
As for taking them out, it's almost always faster to take your fleet through a Transport and take out the Command Station. One can always come back to deal with the guard posts separately. On the hull aspect, they're Refractives and take more damage from Missile Frigates.
If you don't like jetting your entire fleet around in transports, you can try to set up a sweeper team using a decloaker, a cap of Missile frigate-Is, and a transport (a few is better, using one as a decoy). Put them all into one control group, and ask the Missile Frigates to target Mines preferentially. Or just switch to Neinzuls.