Trade in War

It's simple self interest. If you perceive trade as benefiting your interests (which, by definition, would imply not giving your opponent a meaningful edge in the conflict) then you trade.

It's the same thing that decides when wars end. Either side is free to surrender or withdraw at any point in a war. They continue fighting when they think that the result they can obtain (win or lose) in the future will be better than the result they can obtain in the present. They stop fighting when that no longer becomes the case, at least assuming non-insane leadership.

War isn't the end of relations. It's not infrequent that enemies today are allies tomorrow, or vice versa. Nobody ever knows what tomorrow holds, and history seems to love a nice plot twist.

You're ignoring the principal-agent problem. Even when national leaders aren't technically insane, they will often choose to continue fighting well past the point where it could possibly benefit their country because that suits their own personal interests. So they keep sending their citizens to die because there are no personal consequences.