No, the sod cutter will not work. Do not till the Bermuda. Hand dig when the soil moisture is such that the soil will crumble and Bermuda can be separated with minimal cutting of the roots. Fix your soil so that it will support close spacing of plants that will not allow any sun to the bed surface. The shade will hold the Bermuda at bay.

I have converted several areas of bermuda lawn to garden by using round up twice, 6 weeks apart and then covering the area with 10+ sheets of newspaper. Wet the newspaper thoroughly and cover with a few inches of mulch. Keep the newspaper damp if you can. I would hold off planting or disturbing that area for a year.
Clear plastic in the summer probably can be used to kill the grass instead of Roundup but I'm not sure you would get down to all the roots.

Stevega's suggestions are pretty thorough, but at least here in California, it also helps to wait until warm weather, and fertilize and water first and in between applications of Roundup to get better regrowth. Factor in that we don't get any rainfall outside our winter/early spring rainy season, so it is important to "goose" the Bermuda grass into faster regrowth mode.

Using Roundup to kill the bermuda grass is a better method than trying to skim it off with a sod cutter, as you won't begin to get all the deeper rhizomes out, and you'll be back where you started. As already stated, it does take several applications to really get complete kill, and you should wait to do the first until after the grass is actively growing again when it warms up,(probably mid-April in most parts of California, earlier in coastal southern California). It really helps to use water and fertilizer first to get it actively growing, and only spray once air temperatures are at least 65 or 70F for best kill. Immediate kill will be even more evident if air temps are above 75F.

Then you wait a week, then start watering again to encourage regrowth, probably a month to 6 weeks after your first application. You may need to repeat the process a third time for complete kill. I'd suggest using concentrated Roundup, and if you skip any of these steps, you are most likely to miss killing all of it, and have to spot spray with herbicide afterwards. That may not be a problem if you are itching to replant immediately, but it is much harder to control Bermuda grass if it is growing amongst valuable new plants that you don't want to damage.