While one copy of the support is on the board, it should trigger 2 damage when another one is played, followed by the 5+2 damage of the second copy.
Oath of Jace does trigger when in this same scenario so I don’t see why Chandra’s shouldn’t.
Reinforcement doesn’t seem terribly consistent.
Some cards act like you just gave a bonus to attack/toughness and others trigger the whole “summoning effect”.
I have similar issues with some red cards that trigger on colorless, or creature stealing. Sometimes when they get it back, everything triggers, sometimes it doesn’t. It’s consistent with a creature, but not consistent behavior for all creatures.