So out of the box it is not possible to use different search handler through Sitecore SOLR content search. Sitecore's SOLR implementation is based on SOLR.NET, which hard codes the search handler to use in the "SolrQueryExecuter" class.
/// Default Solr query handler
public static readonly string DefaultHandler = "/select";
The DefaultHandler to use from the SolrQueryExecuter class.
The original code can be found here.
You could implement your own executer class an patch it through the SolrNET Startup container you are using.
Register<ISolrQueryExecuter<T>>(c => new SolrQueryExecuter<T>(c.GetInstance<ISolrAbstractResponseParser<T>>(), connection, c.GetInstance<ISolrQuerySerializer>(), c.GetInstance<ISolrFacetQuerySerializer>(), c.GetInstance<ISolrMoreLikeThisHandlerQueryResultsParser<T>>()));
The initialization code for the default SolrQueryExecuter do in the SolrNET.Startup.Init method.
The Original code can be found here:
The SolrNET.Startup.Container is a public property, and thus can be altered.
However this will not give you the option to choose which one to use on query time.