If you drive in the UK, you'll see road signs that have one word on them - STOP - and this is perfectly acceptable. English learners are sometimes taught as a "rule" that a simple sentence must have a subject and a verb as a minimum, but that isn't the case with orders.
Orders, such as on signs and slogans, don't have to follow the grammatical structure of everyday speech. They are often in the imperative mood, aimed at the reader (or hearer) who is the implied subject. In effect it is asking you to obey the order.
"Stop Asian Hate" is punchy and succinct - it seems clear to me that it is a campaign to stop hate against Asian people - what else could it mean? The only other alternative would be that it is a campaign to stop Asian people hating something else - but what? It wouldn't be a very good slogan if there was no object.
Your suggestion of "Stop *anti-Asiananti-Asian Hate" doesn't sound right. It could mean stop hating people who are anti-Asian. Consider "female oppression" - you'd understand that meant oppression of females. WhyYou wouldn't say "anti-female oppression"?.