Part of Pence's, and Indiana's, problem is that the negative reaction to the law isn't just hurting their images, it's costing real money in lost dollars from would-be visitors who instead are or will be boycotting. The specter of similar negative fallout seems to have pushed Arkansas' governor, Asa Hutchinson, to refuse to sign a similar bill into law. Pence and Hutchinson each has called on his state's legislature to "fix" or to "clarify" the legislation so it won't permit discrimination against non-heterosexuals.
I'm skeptical, to put it kindly, that either state can produce fixes that will satisfy critics. Those agitating for these laws have already admitted that their point is to allow private citizens (as business owners and possibly even as employees) to express their disapproval of ... well, just about anything their religion (putatively) deems objectionable. If your religion deems same-sex marriage immoral, for instance, you, the religiously observant business owner, shouldn't have to "participate" in the ceremony. The laws' supporters aren't going to be satisfied unless they're allowed to express their "faith" freely. (Incidentally, they haven't explained how providing the flowers or wedding cake constitutes "participating" in the wedding ceremony.)
But you know, even if these state-level RFRAs (inspired by the misbegotten federal law of the same name enacted during Bill Clinton's administration) were deemed unconstitutional by some mythically impartial Supreme Court (one that doesn't exist right now), it wouldn't stop the zealots. They've told themselves a myth of victimhood and moral probity that is too satisfying to give up.
The truth is that religious belief already is offered a host of protections in our legal system, and always has been. Calling for even greater deference to religion seems like a suspiciously convenient way to excuse facially discriminatory behavior — behavior you couldn't get away with if you claimed it merely as personal belief. You must prove your organized religion actually holds the views you claim, but that's not difficult: federal prisoners have been gaming the religious-belief exemptions in the penal code for years to get special meals, for instance, and they don't have expensive attorneys on call as the pro-RFRA forces do. The Justice Department and the courts would rather err on the side of lenience and permissiveness when deciding whether (putatively) religiously-motivated behavior is the product of genuine belief. (Nobody even wants to think about the impossibility of gauging the sincerity of such belief.)
The "religious freedom" debate stands the cultural and political reality of the United States on its head. No nation in the developed world is more publicly and privately solicitous of organized religion, yet here we are, contemplating giving even more license for religious expression in previously off-limits (or at least conventionally avoided) areas of our civic lives. The unavoidable suspicion is that these laws have a different purpose than the avowed one of defending supposedly embattled religiosity. What they aim at is nothing less than the state's complete surrender of its right and duty to hold religious believers to the same standard under the law as the rest of us.
None of this is why the Indiana law is causing such an uproar. But it should be.