If my Active Pokemon is Togekiss with the "Bright Veil" Ability, do I bring up one of my benched Pokemon when my opponent plays an Escape Rope?

No. Bright Veil prevents your part of the "Each player switches his or her Active Pokemon" part of Escape Rope, so your Active Pokemon stays put and you can't bring up a benched Pokemon. However, your opponent does switch.

Relevant cards