FORT OGLETHORPE, Ga. — Just down the road from where Union troops suffered their worst defeat of the Civil War, Jeff Epperson sang the praises of his congressman, Representative Tom Graves, whose Defund Obamacare Act set the table for the partial government shutdown.

Even though business has been slow at Mr. Epperson’s sword and knife shop since tourists stopped visiting the historic Chickamauga battlefield, which closed on Tuesday because of the furlough of federal workers in the shutdown, he said the only thing that would weaken his support for Mr. Graves would be if the congressman caved in now. In that case, he might vote for a more conservative choice in the next Republican primary.

“If he backs off, then I would say absolutely I’d be inclined to look for someone else,” said Mr. Epperson, whose store flew a Don’t Tread on Me flag.

The Republican insistence in the House on tying financing of the federal government to dismantling the Affordable Care Act is being driven by a deeply conservative caucus from places like Mr. Graves’s 14th Congressional District, newly created by Georgia’s Republican-controlled Legislature.