This is a great example of where the "interlocking" halftones are already there. I really don't suggest doing a "sheet" of white ink under the black... but using two whites of the same screen unless you are going on manual. You see if you just invert the image, there are your white separation halftones perfectly interlocked with the black -- which will lock-out the shirt color no matter what it is. This should really be done on all "simulated" process prints... much better than too much white, and eliminates shirt-color showing through. But you would want to hit the white, (the dots will gain) - then flash, black (those will gain over the white gain and give some grey mix) - flash and maybe hit the white again to bring back some of the black dot-gain which pushed it a bit too dark.
If the dots are big enough the pre-halftone is ok as long as the angle does not conflict with your screens or other chances of moire. It's already separated unless you really wanted to do greyscale -- then you need the original or will lose fidelity but you could "blur" the halftone one until you don't see a pattern anymore and then re-separate with black/grey/white. Interlocking those is also pretty easy with inverting the channels of dots. It's not "experimental" I have been doing interlocked halftone separations and prints for quite a few years now with results unsurpassed when it comes to comparison with other halftone techniques under a given scenario like this one.
You would never want to generate a black and white image with black halftones and white halftones at 22.5 degrees "overprinting" each other with gaps between them, but you can see like I said this is a great example of what 2-color interlocked screens are. The white is already there in halftones for you as well. Doing a big square of white is "in my opinion" the wrong way and produces too much lightening of the black (looks grey) and also too thick of a hand/feel with the sometimes shiny-look of plastisols overprinting. Just my two cents.
