When I specify multiple passes, the software often goes around small holes or other details multiple times in quick succession, which I think is causing extra melting.
Is there any way to force the software to just do the same cutting path 10 times?
You can quickly copy all of the parameters of a layer to another layer by selecting the master layer to replicate, click the button to copy the settings into cache, select the target layer, click the button to apply the cached settings:
Have just your two layers of the design and use Cut Selected Graphics. Then Select all, run the job. After the first run, only select the object on the line layer and run the job.
The caveat to this is understanding Start From and Job Origin. I personally use Start From Current Position for almost everything, so in a case like this I would draw a non-output frame around the project and always have that in my selection to retain object alignment relative to the layers being run.
If you use Absolute coordinates, you would not need the non-output frame.