I think you've just described King to a tee, not Graham.
Graham does portray himself as a Reagan Republican in interviews and then goes and stabs the GOP in the back and vocally attacks the GOP for doing so-and-so, but Graham really hasn't gone on TV and played tough guy conservative warrior since the Clinton impeachment (where he faced King attacking him from the left). Certainly Graham does not have the "image" of a no-nonscene straight shooter in Washington. Graham has faced off against Paulbots on foreign policy, and portrayed himself as much more of a militant hard-liner, which he is, and votes accordingly. One reason why his previous challenger went nowhere in 2008 is that the guy was to the left of Obama on foreign policy and preached the ANSWER crowd talking points.
King, on the other hand, has the phony "conservative warrior" stuff down to a science, which is why he's stayed in power for 15 years. He portrays himself the lone guy in Washington who "understands" the "seriousness" of the terrorist threat, while in private he's had numerous ties and friendships with radical terrorist groups. He routinely trouts out all kinds of dog-and-pony shows when he's up for re-election, from announcing he's having a hearing to "investigate radical Muslims" in the U.S., to working the conservative talk shows (where he is greeted by fawning conservative talking heads) to promise he's going to fight the GOP leadership and go after illegal aliens. In all of these cases, he portrays himself as a tough-as-nails straight shooter who is fighting the GOP leadership for conservative principles. None of these "efforts" he lead to anything or amount to Jack Squat (it's like Donald Trump waiving the birther flag and promising he will "prove" Obama's real place of birth), but it does have the effect of rallying gullible conservatives to his side every election cycle.
That sounds like Grahamnesty and Mccain. Maybe King hasnt shown up on my radar screen because he’s just a single House member versus a Senator.
My favorite Grahmanesty story was ~ 2011 or 2012 when he went on a Sunday show to suggest that we change the US constititution to end birthright citizenship.
All it did was set off a firestorm agauinst Republicans and agaist opposition to immigration reform, and the MSM kept reporting it as ‘...Republicans are talking about...’ Of course Grahamnesty never mentioned it again having done his work.
But what a coincidence, Grahamnesty is one the most pro-amnesty in the congress.