You could change the encoding, for instance define two dummys this way:

         asian  black  white
dummy_1    1      1       0
dummy_2    0      1       0

(there are many other ways). dummy-1 compares white to non-white, and the other compares within the non-white group. This is as if comparing white average to non-white weighted average, with weights takes as sample proportions.

If you want some other weighting, you can use a custom contrast.

You could change the encoding, for instance define two dummys this way:

         asian  black  white
dummy_1    1      1       0
dummy_2    0      1       0

(there are many other ways). dummy-1 compares white to non-white, and the other compares within the non-white group. This is as if comparing white average to non-white weighted average, with weights takes as sample proportions.

If you want some other weighting, you can use a custom contrast.

EDIT You say (in a comment)

I believe the group of managers want to see all the levels in our model not just two

But all the levels are in the model! It is as simple as understanding that $a+b = a+b +0\cdot c$, the "omitted" level is there, but with a coefficient of zero. So maybe this is only a problem of reporting, it might be a good idea to present the result with this implicit zeros made explicit. For some examples (and R code) see

• What to do in a multinomial logistic regression when all levels of DV are of interest?

and

• https://stackoverflow.com/questions/67225238/is-there-a-way-to-change-in-referent-category-in-the-gtsummary-to-ref-or-a