Should I soak it in hot water or steam it to make it more pliable and force it back into place
Unless you are in a position to make a new one I think this has the greatest chance of success. After you get the strip out to replace the cane and are then ready to reinstall it you might be able to coax the piece as-is back into position, starting in the middle and working outwards, or starting at one end and working your way along. But obviously with the wood dry like this, and possibly/likely brittle from weathering, there's a greater chance of the piece cracking on you.
Heating the wood (along with moisture, although that's not vital) is the usual way wood is made pliable enough to bend. I would use boiling water for this, not merely hot. It's probably not necessary to heat the entire piece, I'd bet it'll work to locally heat the right portion(s) to make the strip amenable to squeezing back into position but be guided by how the wood is responding, not by your prior plan. Temporarily clamp/wrapping in rubber strips as you go — it wouldn't be safe to tack the wood down as you go in case you need to start again!
It may be best to get used to the idea that the piece will crack on you at least once even with heating, and be prepared to continue. These strips aren't structural, so it wouldn't be the end of the world if they aren't one long continuous piece of wood, although obviously it won't look as good.
You might be temped to re-use the existing nail holes. I would advise against this unless you're moving up to a thicker gauge of brad/panel pin.
Same size or not, make sure your new nails aren't too long! If you can't find brads in the gauge you want that are short enough don't hesitate to shorten them yourself1.
If you are repositioning your nails pre-drill for each and every one to help ensure against splitting2. The holes can be undersize.
Do ensure you leave the wood to dry out fully before tacking back into place. Although it'll undoubtedly look dry in about an hour or so it would be best to leave it for a few hours, overnight if possible3.
1 Regardless if you use a mini drill and a cutoff wheel, a pliers or wire cutters to shorten the nails do not worry about losing the point. Nails (of all sizes, not just small brads) can be completely blunt and still work well, actually slightly better in some cases!
2 This is assuming you're nailing by hand, disregard if using a nail gun.
3 Overnight drying, or even a full 24 hours, is advisable before continuing work any time you get wood soaking wet. Bear this in mind if you decide you need to use oxalic acid or another product to try to combat the greying, after the thorough rinsing stage.