I can't agree with your thinking. If a player forces arbitration and the club doesn't like it then they may well trade the player. But it's got nothing to do with extensions. A team offers extensions because of perceived economic advantage not out of any sense of loyalty to the player (the only exception might be a end of life type contract for a long time team icon, which is economic in its own way).
If anything this tells me that the Braves have no interest right now in extending Folty. If they did, they would pay the extra $100K as an inducement to try to keep him at the extension negotiation table. Instead the Braves are taking a hard line which tells me that they really don't care what Folty thinks or wants.