It definitely sounds like you’re dealing with more than just a simple clog here, especially given that barely half a gallon of water is causing a full backup in both sink basins. That suggests the obstruction might actually be in the trap itself or immediately downstream of it, rather than 12 feet down the line. A partial clog combined with heavy buildup from fats and oils (which tend to congeal in cooler pipes) can make even small amounts of water behave unpredictably.
 To your question about bypassing the P-trap, technically yes, it would allow water to flow more freely if the trap is the restriction point. But it’s not a long-term fix, and even though you’re only dealing with gray water, omitting the trap could still invite odors, insects, and bacteria back into your kitchen over time. In setups like yours, especially in rural or off-grid homes, some folks do opt to bypass or modify traps, but they usually add some kind of waterless trap alternative or venting system to avoid the smell and hygiene concerns.
 That said, your issue might not be with the trap or the clog alone. It could be an airflow problem, as you suspected. Without proper venting, water can’t move efficiently through the system and will push back up, even if the pipe isn’t fully blocked. A blocked or nonexistent vent can cause symptoms just like a clog.
 If you’re not seeing results with a snake or boiling water, and you’ve already cleared the P-trap manually, it might be time to have a professional evaluate the slope of the pipe and check for hidden buildup or venting issues. If you’re in Singapore or nearby, you could 
click here to get in touch with licensed professionals who specialize in these kinds of drainage problems. Sometimes a camera inspection is the best next step, it saves a lot of guesswork.
 Hope you find a clean solution soon. No one wants to be washing dishes in the bathtub for long!