My take may be wrong, you have been warned! But I'm as certain as anyone can be when you say "side battle" so...
Yes and yes. The weapon goes with Ahimelek and does not go off again, as far as I am aware, and neither do the character abilities.
Foreign Sword was interrupted by Christians Suing Another, so it does not have the opportunity to reactivate until after that ability ends...which includes the side battle, so if it is no longer in battle when the original battle resumes, it does not reactivate.
There is no reason you cannot interrupt/negate a weapon that causes special initiative; it is just like any enhancement, or card for that matter, that would cause SI by removal.
As far as I can tell, this is a legal play, and there would be a side battle where Asahel dies (if no interference), the original battle resumes where Ahimelek returns victorious with Foreign Sword, which reactivates and goes back to removing Babylonian Soldiers from battle, causing Special Initiative again.
