Is there English counterpart(s) to Japanese old saying, “Present salt to your enemy.”?
I would think the idiom we are looking for is closer to Provide him with a fighting chance or a level playing field
As for the Don't kick a man... I would contest that the rival in the narrative might not have been down, just lacking salt... I think the saying reflects something earlier than when the enemy is down (but could possibly also be used in that context)
In addition to phrases such as "play fair" and "don't kick a man when he's down" is the concept of Chivalry.
When used in a modern day context, this entails standards of conduct such as courtesy, generosity, valour and fairness towards one's antagonists.
Originally (and still today, when used in a historical context) Chivalry meant the code of conduct required of a knight in armour - as such, there appears to be a strong parallel with the original question's reference to Bushido.
Based on your historical narrative, it seems that an ideal counterpart would emphasize a spirit of fairness that trumps an opportunity to exploit a weakness during some struggle between two opponents.
That being said, an English saying that I would offer is:
Don't kick him when he's down.
The saying refers to some kind of fistfight between two opponents, with the exhortation that, should you manage to knock your opponent to the ground, you shouldn't start kicking him while he is unable to defend himself.
The phrase is often used in a figurative sense. I managed to find one instance where it was apparently used by a former U.S. president1, 2:
“Don't kick him when he's down,” the president warned. “We can't. We have to be very conscious of Gingrich's standing. He's the only one that can pull it together.”
I think the closest English analogue (at least in the UK) to "bushido" might be "cricket." In English, it's usually used in the negative: "that's not cricket," describing something that gives you an advantage perceived as unfair.
Honor among enemies, seems to me to be exactly what you are looking for.
The word honor has been around for a long time and can mean several things, but I think the most applicable would be:
a : a keen sense of ethical conduct : integrity (a man of honor)
b : a showing of usually merited respect : recognition (pay honor to our founder)
c : one's word given as a guarantee of performance (on my honor, I will be there)
So the phrase honor among enemies literally states that all involved will conduct themselves in such a way that they may be said to have honor(recognition, respect and a guarantee of good conduct) from their opponent.