Good point though I could see him being a cabinet pick for Anderson. Also how could we remove some of the other moderate candidates such as Baker?
I don't know that you really need to. Anderson burst into the front of the pack post-NH. By that point, Baker was gone.
I think the path for Anderson is something like this:
1) Connally invests significantly in Iowa and New Hampshire at the start and also pitches himself as a candidate for the Moral Majority. This isn't actually implausible. Iowa was Connally's original strategy and he was considered the runner-up among Evangelicals. He had a great relationship with Billy Graham.
2) Connally goes on to win Iowa, Bush comes in second, and Reagan comes in third. The Reagan campaign is shocked. The entire race is flipped on its head. If Connally wins Iowa, you've butterflied Reagan's comeback to an extent because there isn't going to be an opportunity for a two-man debate, in which Reagan can repeat his iconic microphone demand. With three serious contenders, there'd be no real impetus for the debate debacle in the first place. Instead, you'd likely have a standard NH debate with the major candidates (Connally, Reagan, Bush, Crane, Baker, Dole, Anderson).
3) Reagan had a lot of structural advantages in NH. His politics made sense for the state's electorate. The state's primary traditionally supports the underdog and resets the narrative of the race. All of this helps Reagan. That said, his campaign will be badly damaged from IA and there's no debate moment ITTL and so you get a scenario where Reagan wins but barely. Bush comes in second. Connally takes third. Now, Dole, Crane, and Baker are essentially "nobodies" in the race. They fade into oblivion as per OTL and the press begins its OTL infatuation with Anderson as it begins to hit Connally with negative stories.
4) South Carolina is the hard part. Your most plausible scenario is for Bush to stay out of South Carolina altogether (which was his original plan) because his campaign won't be as on strong footing as OTL so he'll be less cocky. But that actually hurts Anderson. It's not totally unrealistic for Bush to still compete there, thinking Connally/Reagan will split the conservative vote and he can come up the middle. In fact, if Connally is stronger than OTL he may be more compelled to do this. The money and time spent in SC will not yield dividends for Bush ITTL, just like OTL.
5) Meanwhile, Anderson strikes. He BEATS Bush in MA while the candidate is off throwing away his chances in SC. And he wins Vermont the same day. He came very close in both states IOTL so this isn't a stretch in a weakened Bush scenario. Now, Anderson's won VT and MA. Bush has won zero.
6) Connally wins South Carolina and Florida right afterwards. Now, Reagan's basically dead in the water. But for a lot of voters, Connally isn't a good nominee. He's seen as corrupt and tied to Nixon. Conservatives split between Connally and Reagan. A lot of voters want Bush instead, but they worry he's not capable of winning. Super Tuesday is nearing and he hasn't won a contest. Anderson's to their left, but they'd prefer him to Connally or Reagan. On March 18, Anderson delivers a crushing below to the rest of the field, taking the Illinois primary. Reagan in a distant second. Bush in a distant third.
7) At this point, Reagan and Bush hobble along but neither has actually won. Reagan's planning to go to the convention, but his campaign's basically going to be out of money at this point -- so I don't know how he can. Bush has a big cash advantage, but voters aren't buying what he's selling and are worried about electability. All the money he spends trashing Connally and Reagan actually benefits Anderson more than it does him.
8) In a very messy and exhausting campaign, Reagan and eventually Bush drop and Anderson and Connally are left for a two-man show. Anderson clinches the nomination by taking California's WTA primary. It's a bloodbath at the convention. They try to make it open and get Reagan or even Ford on the ballot in that way, but they're not successful. Bush throws his support behind Anderson, knowing it's his best chance at a cabinet spot. Helms organizes a walk-out and vows to run a third party bid. Anderson probably taps Lugar or Van Der Jagt as his running mate. If he and Laxalt can stomach each other, they maket the most sense in terms of ideological balance but it seems more likely to me Laxalt walks out with Helms. Van Der Jagt could run with Helms, too.