I use a plastic toothpick (Doctor’s Brushpicks, to be exact), on the theory that exerting the upward force with something bendable cushions the upward force on the reed. The toothpick is more flexible at one end than at the other. The paper you’re using sounds good, though the method (which I have also used) may threaten the most vulnerable and unsupportable part of the reed, i.e. where it is attached to the plate. But at least the paper is flexible and distributes the pressure in relation to how you manipulate it.
My usual method has also been to gently repeat the upper motion as you describe. The worst thing, it seems, is bending the reed back and forth at its base, thereby producing metal fatigue. How about instead fixing your apparatus in place with a mild upward tension on the reed and leaving it there to do its work over time?