That’s the thing. Illustrator/Inkscape are designed for precision work, it’s possible to do it in PS, but the right tool makes every job easier. Clipping masks and rasterizing is the way to go.
If GF ever supports masks the workflow will get a bit simpler, but your guess is as good as mine there.
An easy way is to fix the original photo as a layer in Gimp or what you are good with and make that pure white. In inkscape I use the clipping path but just to help center the image and the GFUI complains, but as it will not engrave there anyway i can ignore that,
One way to get the PS precision you’re looking for is with guides but since you have both programs and had already done the work in Illustrator the fastest method I can see would’ve been to open your Illustrator file in PS. It’s similar to rasterising in Illustrator but with more control/options.