Will Rings of Brighthearth + The Chain Veil let me use Chain Veil's ability more than once?
Yes. The sequence would/could be as follows:


You activate The Chain Veil, and put its ability on the stack.
The ring's triggered ability triggers, and is put on the stack.
(after all players pass priority) The triggered ability resolves, you are given the choice to pay 2 and copy the ability, and the copy is placed on the stack.
(players pass priority) The copied effect resolves.
(players pass priority) The original effect resolves.


A ruling from Gatherer confirms that The Chain Veil's ability can be repeated, i.e. each effect allows an additional loyalty ability to be played, on top of the single normal use.


  Each additional time The Chain Veil’s last ability resolves will allow you to activate a loyalty ability of each planeswalker you control an additional time. For example, if you activate The Chain Veil’s last ability, untap it, then activate it again, you can activate a loyalty ability of a planeswalker you control three times that turn.