Now departing from the OP situation to address other questions/observations:
The shooters opponent should be careful (yes, he has the option of allowing the run to continue) of allowing the shooter to remain at the table because the shooter could realize his error and "fix" it.
Yes, there is danger in allowing the run to continue. If the 1-ball is played and the shooter pulls the trigger on the 1-ball, Player A would continue now since the 5-ball foul was not called.
...if the shooter noticed but was trying to get away with something, then it would probably just be a forfeit game, right?...
Possibly, but the only way I see it playing out that way is if the shooter basically admitted to willingly committing the earlier fouls (2-ball, 3-ball, 4-ball,) then fouled on the 5-ball and just turned over BIH on the spot. In that case, yes, I would rule UC.
At least in my quick analysis, that is not at all the same as allowing the shooter that does not know he is fouling to continue. If the latter happened and Player A whined because the previous fouls were not called, my simple response would be that they were at fault from the beginning, so they really have no leg to stand on.
...And if the other player noticed and figured he'd let the shooter run them out a bit before calling foul in order to get more advantageous position, then what would happen?
There is no rule that says that a player (as opposed to a referee) must call a foul when it occurs. Again, - if Player A whined because the previous fouls were not called, my simple response would be that they were at fault from the beginning, so they really have no leg to stand on. You can make an argument that it is not completely unreasonable to consider UC, but there is always a danger, since by doing so you are essentially denying Player B the opportunity to play by the letter of the rules.
Can it be considered a twist on the true intent of the "fouls not called" provisions? Maybe, but still legal nevertheless.
In both of these cases, the results of the referee's interview of the players must be taken into account. If a player gives a clear impression of UC behavior, as opposed to just taking advantage of the loopholes in the rules, then blast away.
...but the shooter quickly steps up and pockets the one ball before his opponent actiually calls the foul...
That would be at the discretion of a referee, but would surely be called UC under the circumstances. The "quick" action is all the evidence you really need of UC intent:
Ref: "And, Player A, how many other times during this event have you dived at a ball with no PSR and fired a shot off in 2.7 seconds?"
Player A: "Uh...."
End of discussion.
Does simply discussing an obvious foul constitute the calling of a foul? What do the rules say about that?
The rules discuss stopping play, which clearly contemplates at least the suspicion of a foul. Play has been stopped, so a resolution has to be achieved before you continue.
Buddy