How to get mDNS working for Chrome on Android?
I don't think so.

The system DNS resolver in Apple's platforms has in-built support for mDNS, so that pretty much any app eventually winds up using the core getaddrinfo function and supporting mDNS in URLs and hostnames and whatnot.

On Android and other platforms, even if there is a Multicast DNS daemon running and libraries available for using it, "typical" DNS lookups do not. So while recent Android releases do give developers some mDNS features as part of the platform, most do not go out of their way (as they would need to do) to use them.

It's a bummer, but AFAIK there's nothing short of rolling your own custom Android build (or convincing Google to change it) that could really solve this.