Backdoor it via Japan. Britain is allied with Japan, and Japan was considered the most likely US geopolitical foe
So interests in Japan provide more support to Filipino Rebels (they did OTL), get caught and lead to a diplomatic crisis, then Russo-Japanese war comes along and the IJN has a shooting incident with the USN that leads to dead Americans. No war but clearly both sides are under the threat of war, US never mediates the end of the war
Anti-Asian racism makes US-Japan relations stay bad, a few war scares, some hate crimes, politicians mouthing off keep things at a boil. The UK tries of course to mediate but dares not drop the Anglo-Japanese alliance with Germany being an immediate threat
WWI actually breaks out, and the US skirts the boundaries of neutrality against Japan. This leads to an incident where the IJN violates US territorial waters to attack a German ship hiding within. That is an act of war, the US demands concessions, Japan refuses, negotiations are slow, then an incident in China occurs, tempers flare, war is declared
At this point the US is not at war with the UK, and the UK is desperately trying to get the US to make peace and pointing out that the AJA means they aren't obligated to help Japan. Over the course of the war however, incidents between the USN and RN occur, tensions over RN actions vis a vis the blockade of Europe mount, already high from the UK causing a recession in 1914 by dumping its US assets (OTL this was stopped, but that is one man's personal decision, easy to change). Eventually after enough incidents, the US declares war on the UK and is cobelligerents with the CP