Hi @edrmorris,
The download API allows you to build more complex query including excluding specific taxa. See this example: Not a bird download - GBIF Data Blog - #2 by mgrosjean.
If you are new to the API, I suggest checking our Data Use Club recorded webinar on the topic. See the video here: Data use club - practical session 3 - recording and resources
You can also check our API beginner blogpost: GBIF API beginners guide - GBIF Data Blog
I hope this helps. Please let us know if you have any question.
Note that there is no way to exclude taxa in the search API nor on the GBIF website.