It is almost literally impossible to "seamlessly" together split sections when engraving by fiber rotary. The simple reason is, the math doesn't work...

The mirrors are incredibly precise in what they do. The rotary, and the part you're engraving however, are not even close. To get a perfect seam every time, the part diameter, in all directions (static dia, rolling radius, azimuth), cannot vary more than say .01mm. Any changes will result in a visible engraved overlap, or visible space between lines. The rotary's themselves aren't bad at accuracy, but they need to adjusted to a higher res than the factory 6400 ppr mine was set at. I have mine set for 20,000. Much smoother transition and less start/stop shock. Note, in the F3 menu, "other" tab, at the top is a place to enter a 'start mark delay', 300ms works pretty good to give the chuck time to stop before the next engraving starts. This might help your current problem somewhat.

Learning to use the 'rotary text mark' feature is a good thing, when engraving text the rotary automatically engraves each letter individually. It also works with individually hatched graphics that may overlap in the rotary plane but don't touch each other; if you can separate the pieces so they'll stay in focus around the radius, just individually hatch the pieces. Great tool for doing that, is the "Hatch one by one" check box in the hatch menu, that's what it's for one note on hatching one by one, if any graphics you consider 'one' graphic has holes in it, those graphics will need to be grouped first, then just batch-hatch all graphics one by one.

hope this helps--