You can make this even simpler if you export one frame of this logo into photoshop, save it as tif, put a second layer under it, erase everything in the bottom layer, save as tif with layers rpeserved, or as photoshop psd file with alyes preserved.
Import this still into final cut and save it to the project.
Double-click the thing, set a two-second clip duration, and now in the “motion” tab, you have some keyframe controls. The keyframing is just like what you are used to in 3-D programs, more or less. You can use a rotation command to make the logo edge-on, set a keyframe, advance two seconds, turn the logo 180 degrees to reach the opposite edge, set the last keyframe, done. Now just select that clip, and copy-paste it to the timeline as many times as you like to make it loop.
If you forgot to save an alpha channel, you can still make this work, by changing the blend mode of the clip, go to Modify>Blending Modes, and try the options there. You’ll have to adjust opacity in the motion tab for some of them to work well.
You can also do this project with more ease and precision in Apple Motion, and generate a rendered looping clip that’s long as you’ll ever need it to be…
You are literally drowning in various methods to achieve this effect. Pick one and try it out. They are all fun.