You don't negotiate. That leaves fate to someone else whose interests most likely don't align with your own. You make your own luck.

State Departments are powerful, but politically complicated and slow. Cash in the right hands goes a long way in many places, and is much quicker - valuable when time isn't your ally.

It's not as simple as you suggest. Hearing someone is imprisoned, or kidnapped, is probably one of the last things that a local consulate or embassy wants to deal with. Unless there is broader geopolitical concern, there is little reason to think that these folks will do much beyond the minimum to keep you alive. There is a good chance that the local diplomatics aren't too sympathetic either because know what you should have known about the country before you got yourself into trouble.

If they think there is some injustice, then it depends on how much it'll play out in the media. The government posture at home will also play a part. You could see the Trump administration playing hardball in a lot of scenarios to demonstrate how tough they are, but then they undermine the general ability of the State Department to operate, so lower visibility problems struggle for attention and resources. The Obama administration provided more resources, but probably wasn't as hawkish.