Old maps of the Americas sometimes had a fictionalized Chinese colony located in the Pacific Northwest. Probably because it was in the middle of no man's land between the British and Spanish claims. Or perhaps it was placed there by advocates for the exploration of the Northwest Passage.
To met your requirement, have China open up by the time of McCartney's embassy. They set up an experimental colony on the mouth of the Columbia river. The old mythical name of Fusang is resurrected from ancient manuscripts. This project, like Russian Alaska, proves too ambitious to maintain and is eventually sold to the US. Fusang is the name in lieu of Washington state.