That iron probably wants 120V to get full tolerable
power (i.e. without burning itself up). But that is all
tied up with the rest of the thermal path. Putting
12V across it will net you something more than 1/100
the power (because resistance goes up w/ temp, so
somewhat self regulating.
You might take the mechanical suggestion about
the tensioning weight, and go further by making the
weight, be a bolt/washer/nut clamp onto the wire
that also is one end of the power feed - position
the feed-weight along the wire length such that you
get the desired power / temperature.
Now, you have to consider whether this hot wire
will nicely bend, or simply melt into, the workpiece.
A concentrated heat source is not your friend really.
It can easily fet away from you. A larger diameter
rod (like, say, stainless welding rod, available in
1/32" up to 1/4") might be more the ticket, and
develop more power in the heater than NiCr sized
for a higher voltage. Particularly if your workpiece
is larger than the length of NiCr you've got, or the
length that develops useful heat at 12V.
Car battery, fat cables, and bolt-clamped welding
rods of varying diameter is an easy experiment to
try. Maybe use a tension spring since the rod can't
just be draped over something and weighted.
But don't discount other methods such as an IR
lamp or heat gun, and a passive forming jig. These
can be a lot more controllable. You can get a heat
gun that's more than adequate for about ten bucks
at Harbor Freight.
PC power supplies can throw decent current, not as
much as a car battery but at least 100-150W out the
5V and 12V outputs. This might play well with the
thinner welding rods (I like about 60A in a 3/32" steel
rod for arc welding, you'd like to be maybe 1/10 or
less I expect, so not-insane). Your length of rod sets
the heat, may as well play around some and see.