The premise is whether there's room enough in the Republican middle for Carter to mount a winning primary campaign. His money quote (and it would be a campaign theme, I'd think, should he run because it suggests blind calculation and lack of principle on the part of opponents):
“It’s not worth it to me to go out and be someone I’m not to win anything,” Carter said.
The notion that Carter is a centrist is the best indication there is about the extreme orientation of the Republican Party of Arkansas (and all the other Deep South Republican states). Carter advocated Medicaid expansion. That's enough to make him poison to the Tea Party. But otherwise, he 1) supported the gun agenda slavishly, with the exception of open carry, a bit of nuttery too far for many others; 2) he voted down the line on the anti-woman abortion bills; 3) he engineered a bodacious tax cut for multi-millionaires; 4) he gave the definitive vote on constitutionally suspect legislation aimed at cutting down black voting strength in Arkansas, 5) he waved through an anti-gay resolution. Only to Republican contenders Asa Hutchinson and Curtis Coleman is this the voting record of a centrist. On the other hand: Are GOP legislators John Burris, David Sanders and Jonathan Dismang centrists? They are widely credited, after all, with pushing Obamacare through the legislature with majority Republican support. Maybe they could form a Centrist Republicans for Davy organization. (PS: I meant this tongue in cheek.)
In the Republican context of today, centrist mostly means personally pleasant. These "centrists" don't rub it in gleefully (see your average gun nut) when they cram their ultraconservative agenda down the throats of middle-of-the-roaders and liberals.