So far, top House leaders have tried to head off such a showdown, but Rep. Mark Meadows, R-N.C., collected signatures on a letter sent last week imploring party bosses to use the government-funding showdown to cripple the health law they strongly oppose.
Bishop is one of 80 Republicans and the only Utahn to sign the letter, though one of his top advisers notes that the Meadows letter doesn't include a vow to oppose any budget deal that includes Obamacare funding.
While Lee is looking for the House to lead out, Bishop says the House has already voted more than 40 times to dismantle or repeal the new health law.
"There is overwhelming consensus that Obamacare needs to be repealed and replaced," he said. "The House has already acted to repeal it, and now it's time for the Senate to take action."
Lee believes the best chance to do so is to pit funding for Obamacare against funding for the rest of the government. He is making this pitch in town hall meetings in Utah, while supporters such as Sen. Ted Cruz, R-Texas, and Heritage Foundation President Jim DeMint travel through the South holding rallies.
They plan to hold a similar event in Washington, D.C., when Congress returns from its recess in the second week of September.
But many in the Republican Party are skeptical, if not downright hostile, to Lee's strategy. They doubt whether it could be effective, since a big portion of the funding for the law is not controlled by the budget bill, but, like Social Security and Medicare, is mandatory. Many Republicans question whether it is practical, since the Democrats control the Senate and the law is President Barack Obama's signature domestic achievement. And they dispute that it is good politics, worrying that the move would allow Democrats to blame any shutdown on Republicans.
Sen. Orrin Hatch, R-Utah, Senate Minority Leader Mitch McConnell, R-Ky., and others including Sens. Tom Coburn of Oklahoma, John Cornyn of Texas and Ron Johnson of Wisconsin have all criticized the effort.
And there are plenty of House Republicans who are skeptical of the effort as well. Rep. Jason Chaffetz, R-Utah, is one of a growing group who would rather try to delay implementation of the law a year than to flirt with a shutdown.
Lee's effort, with a total of 14 Senate supporters, appears likely to fall far short. It would take 41 votes in the Senate to block a budget bill, and it would take 60 to overcome Democratic objections to stripping funding from Obamacare.
But Lee isn't giving up and he looks at the Meadows letter, which Bishop signed, as progress. And while it doesn't come with the same threat, it does show that some conservatives agree with his aims.
"Members either vote to support Obamacare or they vote to defund Obamacare," said Lee spokesman Brian Phillips. "And whether they did or did not sign the letter will be secondary."