You're right, the Dataverse Search API doesn't support POST. Maybe it should. Please feel free to create an issue about this.
The good news is that Solr itself does support POST*. For example, This works fine on my machine (searching the "name" field for "foobar"):
(In curl, "-d" does a POST automatically but above I put in the "-X POST" for clarity.)
I'm not sure what a good workaround would be. If you have access to the Solr server itself, you could query it with POST directly.
I hope this helps,
Phil