Given the outcry against the Chinese (even if the phrase 'Yellow Peril' hadn't been coined yet, or had it?) I can't see any believable way for Chinese immigrants to be treated like whites.
Not treated like whites, but I can think of a few ways to drastically improve the treatment of Chinese immigrants in the United States.
1. Reduce white labor tensions. Most anti-Chinese sentiment and legislation was born out of the industrial struggles of the late 19th century. Chinese were an easy scapegoat for industrial and union bosses that couldn't or wouldn't provide results demanded of them by white workers.
2. Reduce the amount of Chinese immigrants. If there aren't very many of them, they won't be perceived as a threat. Historically Asian immigrants were treated like whites in the U.S. South, as there simply weren't many compared to the large black population. America was a black-white binary from Bacon's Rebellion onward, but the perception of Asians is likely to be more malleable.
3. Have the Chinese government be powerful enough to lobby on behalf of Chinese immigrants. Though a stronger China would also prevent it from being a huge source of dirt cheap labor. You'd probably see the demographic profile being a lot more similar to Japanese Americans --- mostly middle class and literate.
I think a big change of a later Chinese Exclusion Act would be on the demographics of Hawaii. It would be Chinese-dominated, akin to a giant Christmas Island, rather than a hodge-podge of groups with Japanese being the prestige culture like OTL.