I was watching The Butler, and it takes place post World War II but they make no mention of Asian Americans.
Questions: What was life like for Asian Americans before World War II? What was life like for Asian Americans after World War II? Were Asian Americans discriminated against the same way African Americans were before, during and after the Civil Rights movement? Were Asian Americans the target of the KKK and lynching as well? Were Asian Americans allowed to go to white schools and vote? Were Asian Americans treated differently based on whether they were East Asian (i.e. Chinese) or South Asian (i.e. Indian)?
Before WW2? It depends on who you are talking about. The Japanese Americans did fairly well for themselves, while the Chinese Americans were very marginalized by policies ranging from immigration to marriage. See the Chinese Exclusion Act for the draconian policies towards Chinese immigration.
Keep in mind that prior to WW2, immigration from Asian countries didn't really exist outside of those from Japan and so outside of the Japanese and Chinese, you didn't see a ton of wide-ranging policies targeting East vs. South Asians.
After WW2, the Japanese suffered a lot of racism as they were hated from WW2 (bombing Pearl Harbor, atrocities, etc.). That being said, their exemplary service in WW2, notably in the 442nd/100th RCT, also gave them quite some respect as the anti-Japanese hysteria died down. China, having been an ally of the United States in WW2, also saw many anti-Chinese policies lifted and immigration restrictions loosened.
Likewise, Koreans from the Korean War began entering the US and were seen as friends in the struggle against communism. And don't discount the fact that many US servicemen were stationed in Asia after the war, married local women, and saw an increase in population from them.
Ultimately, the floodgates of immigration opened to the US from Asia with the Civil Rights Act and the reforms to immigration made in that time period.
As far as how they were treated with regards to segregation and the South... truth is, Asian Americans were always in an awkward kind of situation. They definitely weren't white, but many times they were given "honorary" white status and allowed in white establishments. This was notable in South Africa's apartheid too where Japanese were considered white and eventually Chinese there became it too. A lot of it was politically driven -- in fact, an example of this was in the DC area where black ambassadors or VIPs from the Middle East or Africa were most definitely allowed to stay and eat in white establishments, but local blacks were barred.
You should check out Ansel Adams' Born Free and Equal. A scanned copy is available on loc.gov.
Adams is very sympathetic to the unjust internment of Americans of Japanese descent. His intro admits an attempt to have the reader not fall into the trap of considering all Asians as one minority group, and asks the reader to consider the individual impact of racism.
It's very compelling and obviously features some stunning photos from Manzanar.