Hmmm, I don’t use that tool much, but I’ve never seen that. I downloaded the svg and tried a few things. Not sure what’s happening. Hint: if you add the svg files to a zip file it’s easier for other people to grab them.
When I selected the individual puzzle pieces and turned the fill color off there is some line overlapping. If all you wanted were the lines you may want to clean those up.
One idea I have, inspired by the Tile Etching thread:
Is when you get the pattern inside the heart to work, turn off the lines and only use the fills. Engrave a color and then sharpie it. Give it a few minutes and engrave another group of puzzle pieces, color, repeat. You do this by using different fill colors for the puzzle pieces and use of the ignore option in the GFUI. There are other methods. The point is you would get colored puzzle pieces inside a heart (on a white tile.)