Now thats a very interesting job. And fun too
To get your lines to line-up on both sides, heres something you could try:
Using the outline vector that you used to cut the shapes, open it in Lasercut.
Using Lasercut, draw the lines where they need to be, I'd recommend you orientate the image so the lines are left to right.
Now select all the lines with the pick tool and assign a different layer colour for them. Set your speed/power.
In the output list top right scroll right to reveal the output check box and uncheck the output for the main shape. Send job to laser.
Place a piece of ply on the table. Use the clip box at say speed 70, power 15 and execute.
This will draw a box on the ply. Set your workpiece in the box and engrave. Only the lines will output.
Back to lasercut, Select the whole image and flip it.
Repeat the above steps. You can now flip and align your workpiece in the clip box. Engrave the lines.
They should all be lined up for you.
Hope it works out for you.
Regards
Dave.