you can have them test serum zinc and serum magnesium to start - these are things that work closely with d3 and are known to be low in ms. the rationale should be that 1) research shows levels of both are low in ms, and 2) because of their interconnectedness with vitamin d3 (which has already been demonstrated to be low in your case AND which is causing unusual reactions when supplemented) that all three should be monitored to insure proper interactions and absorption.
if they'll go for it, test serum copper and serum ferritin also. esp. if you begin supplementing zinc.
be clear that you do not want to test erythrocyte or RBC values, but serum specifically.
the sooner you have results the sooner you can go forward. well actually, you can basically assume that your zinc and mag are low and just go ahead and supplement - BUT, having the numbers before and after you start supplementing is good data to have in your records. once you have copper and ferritin data then you can make decisions about how to roll out zinc supplementing over the longer term.
bahaha world without zinc https://www.youtube.com/watch?v=U1iCZpFMYd0