Pitcher A allows the potential lead runner to reach base in a tie game. Pitcher B comes in to relieve and gives up a hit, then gets the lead runner out on a fielder's choice. Now, pitcher A has nobody left he put on base. Pitcher B gives up a run-scoring hit and the batted Pitcher B put on with a single scores the winning run. Who gets the loss?
As you've stated pitcher A gets the loss. Here is the rule from the MLB official rule book: 9.16(g)
When pitchers are changed during an inning, the official scorer shall not charge the relief pitcher with any run (earned or unearned) scored by a runner who was on base at the time such relief pitcher entered the game, nor for runs scored by any runner who reaches base on a fielder’s choice that puts out a runner left on base by any preceding pitcher.
That's because when Pitcher B comes into the game with a runner on and gives up a fielder's choice. He has put his team in a better position to win the game, not a worse one. the other team is less likely to score a run with an additional out and a runner on than they would be with two runners on.
Just because Pitcher B faced the runner now on base. It's still Pitcher A's responsibility that there are runners on at all. Pitcher B should not be penalized for doing his job (getting an out). If the runner scores it's still the responsibility of Pitcher A.
Case 3 under the comment handles a situation similar to this:
Peter is pitching. Abel reaches first base on a base on balls. Roger relieves Peter. Baker singles, advancing Abel to third base. Charlie grounds to short, with Abel out at home plate and Baker advancing to second base. Daniel flies out. Edward singles, scoring Baker. Baker’s run is charged to Peter
Lastly, we have to think of the intent of the rule, which MLB makes clear in the comment below (H/T to Joe for the suggestion to include this):
It is the intent of Rule 9.16(g) (Rule 10.16(g)) to charge each pitcher with the number of runners he put on base, rather than with the individual runners. When a pitcher puts runners on base and is relieved, such pitcher shall be charged with all runs subsequently scored up to and including the number of runners such pitcher left on base when such pitcher left the game, unless such runners are put out without action by the batter.