The 1/2" is more probably a nominal 5/8" (5/8" before sanded), as 1/2" ply never met US building codes. Adding another layer, keep in mind that the grain must run across the joists, not along them for maximum strength. It can be hard to screw a second layer to a thin one under it without stripping the screws out - you may need to predrill pilot holes in the first layer should you choose to add that second layer. Depending on what you're planning for the rest of the room, that thicker subfloor there may mean the tub edge will be hard to make look good.
To me, it would sort of depend on what the bottom of the tub looked like. If it had only a few small feet, I'd want more subflooring, but if it's fairly flat or has lots of support area, and based on their installation instructions saying it will work, I may not add more. Is the new tub going in the same location as the old one? If it worked, the new one should. If the tub bottom isn't flat or the tub doesn't sit level, you may want to embed it in mortar which will make it feel more substantial and spread the load out.