You'll need to get some Flitz Polishing Compound. Do the entire rifle on the outside AND inside the barrel/Breech plug area. This will keep the staining off. Mine had some of that brown stuff to when i first bought it but the Flitz took it off. Cant guarantee that it will take yours off but it Will prevent any more from ruining the finish.
Dirty from a day at the range,

A simple Windex rag cleaned it right off,
What you may also have at the end of the barrel what looks to be rust/pitting is Carbon build up from BH209, ive seen a few TC Triump owners complaining about Pitting that turned out to be just carbon from that powder.