Feature Request: Support POST when using Search API to enable longer queries

Overview of the Feature Request
Currently, the Search API only supports GET requests. If you try to run a search with more than ~1500 characters, the reply is {"status":"ERROR","message":"Search Syntax Error: Error from server at http://localhost:8983/solr/collection1: Expected mime type application/octet-stream but got text/html. <h1>Bad Message 414</h1><pre>reason: URI Too Long</pre>"} . In order to support longer search queries, Dataverse Search API should accept POST requests.

What kind of user is the feature intended for?
API User

What inspired the request?
Within my institution, a user wanted to create a search that looked for records by several hundred different people. We found that we could only search for about 35 people at a time before hitting the URI length limit. I asked about this on dataverse-community here and @pdurbin suggested I file an issue..

What existing behavior do you want changed?
Search API should accept POST requests in addition to GET requets

Start by locating the Search API entry point and the tests covering its current GET behavior. Reproduce a query exceeding the URI length limit, then verify that equivalent POST requests are accepted for longer queries while existing GET requests continue to work.