I think, the problem here is that one party can impose almost arbitrary costs on the other at little cost to themselves (this is actually my definition of "weapon").

The appropriate way of dealing with the problem is allocating costs where they belong: as long as the requesting party bears all the costs of discovery, they will behave. Or am I missing something?

Yes, allocating costs where they belong would help. In legal cases, we have some trouble identifying where they belong until the final judgement. At that point, the losing side should pay the costs, and this is how it works in UK, AU, etc.

In the USA, legal costs are always own-pay. As I understand it, USA judges have the ability to pass the winner's costs to the losing side, but they never do. I would suggest that costs are always misallocated in the USA, but perhaps others could comment?

This is a classic example of why multinationals should ONLY be incorporating in off shore jurisdictions