Fielders Choice - Earned Run

Question
Runners on 2nd and third, no outs (walk, single, lead runner to third, trail advances on DI on next pitch).  Batter bunts.  Out could have been recorded at first, but pitcher threw home to check runners.  With bases loaded and no outs, next batter strikes out.  Next batter singles, scoring two runs and moving batter-runner who reached on fielder's choice to second.  A WP moves both runners up.  Next batter grounds out and runner who reached on FC scores, with trail runner taking third.  With two out, trail runner scores on a PB.  Batter then grounds out to end inning.

Is the run carried home by the runner who reached on FC an earned or unearned run?

Answer
The defense made the choice to hold the run and allow the batter to reach first.  There should have been one out.  

When three outs should have been recorded, there are no more earned runs that inning.  More runs scored in this inning than should have due to the defenses choice.