Extending Framing Studs From 8' to 9'

I would honestly bite the bullet and get longer studs. Even if the walls aren't structural (which I assume they are not, as it's a basement finishing), you're introducing a lot of extra work (you're basically building the wall structure twice, with the second "wall" being made out of cut to length studs - an extra step for each stud). Also, it would leave you with a lifetime supply of 2x4 for other DIY and woodworking projects.

If you're intent on extending an 8' wall to be 9' tall and worried about unevenness when putting up drywall, the best solution would be furring strips (simply 1x2 pieces of wood overlaid on the studs). This will allow you to both escape the bulk the strapping adds as well as give you the opportunity to shim out the furring strips to get a very straight surface for the drywall. Obviously this is yet another outlay of material and time, so I would still advise getting the right length studs.

I did something similar to this when I raised my kitchen ceiling. I had a recessed area in the middle with fluorescent lights and those plastic clear panels that I wanted to get rid of. It was a lot of work but it saved me from having to replace all the drywall and I didn't need/use those strapping bars. In your case since the drywall isn't installed, I'd swallow my pride and get new studs. You will save yourself so much labor and aggravation.

The straps aren't all that thick to where it should be a concern of messing with the drywall. I went to the website of a local big box hardware store to look at how thick these straps were and found that they are 20 gauge. Another internet search tells me that 20 gauge is 0.0375 inches. That's not going to mess with the drywall. Steel is very strong under tension, these straps are built for this and used often.

It's thin enough that you might be able to put a drywall nail or screw through it without much extra effort, even so it might save you some effort to plan your drywall hanging to avoid needing to put any fasteners through them. There will be holes pre-drilled in the straps for nailing them to the studs but you aren't going to be able to see them through the drywall. Hitting those holes consistently with your drywall nail will take plenty of luck, skill, or both.

Given that the straps are listed at about a buck a piece then it could save you further insult to your injury to find out how many you actually need to be safe and sturdy so you aren't buying more than you need. The lateral loads on the wall will be minimal so it won't take much to keep everything vertical. Even so don't expect the drywall to keep the wall straight. Drywall is not all that strong under tension.

I'd use the straps made for this exact situation rather than trying to come up with some unproven alternative. Go with the pros on this one. Messing this up will make a bigger mess than an lump in the drywall that nobody will see unless they know what to look for, squint their eyes, and catch it in just the right light.