Would Alabama elect a Democrat over a Republican if this was the case

[Fuzzy Bear]

Such an R would have a tough time making it through the primary.  But, no, the R would win, absent a significant scandal.

Conservative Democrats lasted so long in the South because of the "seniority system".  A Democrat could be as conservative as they wanted/needed to be from the South and still rise to a committee chair (which meant money for the state/district) as long as they voted with the Democrats to organize the House.  All of that changed in the mid-1970s, when the Democratic caucus dumped three (3) Southern Democratic committee chairs (Wright Patman (D-TX) of the Banking and Finance Committee, Bob Poage (D-TX) of agriculture, and F. Edward Hebert (D-LA) of Armed Services).   This was the start of the end of the seniority system in Congress; from the 1970s on out, committee chairs had to be somewhat responsive to the Democratic caucus. 

That move had significant consequences.  One consequence was to move a number of conservative Democratic chairs to the center of the party.  Conservatives like Rep. Jamie Whitten (D-MS) who chaired Rules, and Sen John Stennis (D-MS) who chaired Armed Services, voted like moderates for the rest of their tenure (and endorsed the national ticket to boot).  Other conservative "Bourbon" Democrats retired.  What did happen in the mid-1970s is that young conservatives stopped joining the Democratic Party; they all took their chances and became Republicans.  Let's look at the conservatives in the Democratic party who were elected after 1972:

Rep. Larry McDonald (D-GA) - died in a plane crash in 1983
Rep. Andy Ireland (D-FL) - switched to the GOP in 1984
Rep. Richard Shelby (D-AL) - got elected to the Senate as a D in 1986, moderated his record, switched to the GOP in 1995
Rep. Phil Gramm (D-TX) - resigned in 1981, ran for his vacant seat as an R and won
Rep. Kent Hance (D-TX) - was defeated in a Senate primary in 1984, then switched to the GOP and won statewide office
Rep. Ralph Hall (D-TX) - routinely declined to endorse the national ticket, switched to the GOP in 2004
Rep. Charles Stenholm (D-TX) - remained a D until his 2004 defeat.  Voted for Clinton's impeachment, but moderated his record somewhat
 
These are folks who had honestly conservative records at the start of their careers.  There were other Democrats who would be considered "conservative" today that were really moderates, who moderated their record to stay in line with the caucus.  Most of these folks were defeated once people found out that they were voting against liberal legislation on the floor of the House/Senate, but voting in committee to advance such legislation.  (People found out about this in the 1990s due to the proliferation of cable TV and C-SPAN.) 

The parties are too ideological and regimented today for the scenario proposed to happen, absent some unusual factor.