DPI = LPI * 2
or
LPI = DPI / 2
So it would make sense to raster at 2x the planned LPI. I try to do this if I am planning on engraving photographic elements. For solid graphics I would stay at my default of 300 dpi if I were to rasterize them.
Now, that only applies to raster art, which will be for engraving.
For cutting you need vector lines. vector lines don’t care what DPI you select for output to PDF.
Photoshop has some limited vector capability. I wrote a thing about it a while back:
If you have access to illustrator, I would highly suggest playing around with it while you wait for your GF to arrive. If you don’t have access to Illustrator, go check out Inkscape (free).
You will get so much more out of your GF if you have a vector-editor as part of your toolkit.