A great way to force an AI civ to declare war on you is to tie a luxury or resource import to a peace treaty. And then to go and to pillage or otherwise cut the trade route. You need the following ingredients to pull it off:
(1) a victim who has surplus luxes or resources; it is important that you buy the resource, selling one of your own will not work.
(2) a renegotiatable peace treaty
(3) a trade route that can easily be cut (tearing out a few roads or selling a harbour).
The results are stunning. You are fully responsible for engineering a war, but it is not you who takes the blame. Your victim takes that blame, along with a rep hit. You on the other hand are even rewarded by getting War Happyness.
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And don't worry about rep hits from just DoWing too much. It is not required to not have any units in enemy territory to come away with a clean rep after a DoW. In my experience you won't even take a rep hit if you've had units in enemy territory at the beginning of the turn.
If you don't have any trade agreements with America (no current deals) then you should be able to declare war with no rep hit. Unless, (and this is where it gets tricky) a) you have a current gold-per-turn or goods-per-turn (luxuries or resources) deal b) that you are paying c) to a third civ and d) that trade route can only be traced to the third civ by going through America.
e) your trading partner is also at war with America. (Else the route stays open)
If all that is true, declaring war will give you a trade-rep hit, since it will break the trade route to the third civ.
Yes, that would be logical.

That does not mean that it is necessarily true. There are odd cases where you find that, while the deal does show up in the "active deals" tab, it is greyed out. And that, while the money does change hands, the resource/lux does not.
This definitely happens if you have the 'glorious' idea* of signing up one of your trading partners (gpt <-> lux) to an alliance against such a civ-in-the-middle. As long as you alone were at war with the civ-in-the-middle the route was open. Now that your trading partners is at war with that civ as well it no longer is. But does that end the deal? Hmm, let's see what Hamlet would say:
There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy.
* Not me, of course. Ever.