This situation is so narrow and unusual that it is not addressed by the Infraction Procedure Guide, and the handling would be up to the judge's discretion.

State-triggered abilities are unusual, and the IPG does not specifically address them at all. In addition, countering a state-triggered ability never makes sense in normal play except for very rare cases where the spell or ability also has another relevant effect. A judge could reasonably rule that by the letter of the IPG, only one instance of the ability is put on the stack, but a judge could also reasonably rule that the ability would have triggered again, so another instance goes on the stack. Personally, I would expect the latter, because in this situation Player A is clearly taking advantage of the details of a remedy to their own error to gain an advantage in a way that would not be possible in normal play.

Note that no matter what the final outcome is, Player A should be penalized for the initial error here, as specified in this passage from the IPG:

If the triggered ability is usually considered detrimental for the controlling player and they own the card responsible for the existence of the trigger, the penalty is a Warning.

This situation is so narrow and unusual that it is not addressed by the Infraction Procedure Guide, and the handling would be up to the judge's discretion.

State-triggered abilities are unusual, and detrimental state-triggered abilities are even rarer, and the IPG does not specifically address them at all. In addition, countering a state-triggered ability never makes sense in normal play except for very rare cases where the spell or ability also has another relevant effect. 

A judge could reasonably rule that by the letter of the IPG, only one instance of the ability is put on the stack, but a judge could also reasonably rule that the ability would have triggered again, so another instance goes on the stack. Personally, I would expect the latter, because in this situation Player A is clearly taking advantage of the details of a remedy to their own error to gain an advantage in a way that would not be possible in normal play.

Note that no matter what the final outcome is, Player A should be penalized for the initial error here, as specified in this passage from the IPG:

If the triggered ability is usually considered detrimental for the controlling player and they own the card responsible for the existence of the trigger, the penalty is a Warning.