Editing react-native-tcp's android/build.gradle like so fixes the build issue:

repositories {
    maven {
        // All of React Native (JS, Obj-C sources, Android binaries) is installed from npm
        url "$projectDir/../node_modules/react-native/android"
    }
    mavenCentral()
}

dependencies {
    implementation 'com.facebook.react:react-native:+'
    implementation 'com.koushikdutta.async:androidasync:2.1.6'
}

EDIT: Looks like this is patched out of the box in the latest version of react-native-tcp.

As for enabling New Architecture, it just depends on the sizable Android NDK, causes node_modules/react-native/ReactAndroid/build to swell massively, and doesn't deliver any improvements at the moment, so this is on the backburner.