I've looked for, and can't find an anwser to this question: When you declare the attack, is it required to be able to trace the LOS to the target, or can it be traced after using the extra movement points?

When you declare an attack, you must complete the attack before doing anything else. Part of resolving the attack is determining whether or not you have LoS to your intended target (protip: the attack will fail if you don't.) Therefore yes, assuming you actually want the attack to work, you need to have LoS at the time that you declare it.

I think the essence of the problem here is: Can we, while using Running Shot, delcare the target for the attack, after spending the additional movement points?