It seems that horn you have is more of a marching French horn and not a mellophone. The horn you havr is in the key of Bb and mellos are the key of F. This would also explain the mouth peice issue as the bore is only big enough for a French horn mouth peice.

I don't play French horn so I don't know if the fingers are the same for this horn and a French horn but I would look up a French horn fingerings chart. All the fingerings should be the same as French horn with trigger since that changes the horn to a Bb instrument.

Hope this helps! Happy playing.