how do I get the SVG to “know” that a line is covered by a shape
the short answer is, you can not. It is not really a shape, the way its been drawn, its there as a logical path, under the fill in your SVG.
(In LightBurn) You’d either need to delete that section using the node tools, or change its colour to something different and mark that as a layer to ignore in the cuts/layers window.
Interesting, I had a look at your file out of curiosity, and did not have too much trouble getting the result I presume you were after in LightBurn by using the boolean weld (The new Ctrl+B boolean assistant is useful here);
I also noticed the issue you had with the trim tool (I will raise a ticket for this in our support system),in the meantime, if you have this issue again, I have found it is possible to “help” the trim tool find elusive intersections by using the line tool to draw a temporary line with corners/vertices placed at the intersections you want the trim tool to find. eg see the red line (I have zig-zagged the line just for emphasis);
As for your Y axis, check your “Enable rotary” is turned off, or if there is a physical switch somewhere that it hasn’t been bumped on.
Hold the phone!, when I went to retest the Edit Nodes “T” trim on the “3 dairy box.ai” I had trouble reproducing the problem I thought I had, and I think it’s because I did not fully ungroup all the involved paths (there were groups within a group) and looking at one of your screenshots above @sethgodin I think this was causing you issues also. Anyway, I think I’ll leave my note about using the line tool as it has helped me in the past with other weird trim situations.