According to the latest news, a new agreement between Russia and Belarus will be signed in Q1 2018. A mutual recognition of visas was discussed. If that really happens, then you probably will be able to travel to Kaliningrad via Minsk with only one visa, Belarus or Russian one. No details published yet, no idea how it will affect Football Fans, Fan IDs and free trains for the FIFA World Cup. But let’s hope for the best and wait for good news to come. Belarus is an interesting tourist destination. Also, this will allow a lot of additional possible flight, train or bus connections.
UPD: Check the latest update on the visa agreement between Russia and Belarus.