is very useful.
The 'bridge' as you call it is the subfame, and yes, it has to be lowered to get the 'stabilizer bar' (AKA anti roll bar ARB) out.
Replaced mine recently with an Eibach ARB, used mostly the same process as in the link above, but was also doing the shocks and springs so had to do a bit more work.
Undo all the bolts along the rear edge of the subfame BUT NOT the 8 lower wishbone bolts, unless you're planning on replacing those too. Leave the two very long bolts that pass through the 'wings' of the subfame. Not sure whether the 156 is the same as the GT but there's a pipe bolted to the inner of one of the subframe wings, make sure you undo that before lowering the subframe.
When you're happy that only those two bolts are holding the subfame up, get a trolley jack under it as show in the link above, loosen the two long bolts and lower the jack at the same time. If you're only doing the ARB I don't think the bolts have to be fully removed just loosened far enough, and the ARB lowered just enough, to wriggle the ARB out.
The hardest part of the job is getting the subframe to line up again after you're done.
All the best