For me (in inkscape) I’d use some clever path management to join my pieces and then overlay a very thin sprue-width rectangular path or two that crosses the whole piece. From there it is a simple matter of booleans to break the path where they intersect and manually remove the resulting sprue segments. Which Boolean will work best will depend a great deal on your specific design, but offhand cut path or divide in inkscape sound the most likely. Both of those actions have equivalents in illustrator.
If none of that makes sense, I’d start by searching booleans on the forum, the concept is the same across all vector software, you just need to know the specific terms for illustrator.