UDisks2 interface doesn't allow to use DBus.Properties interface

No. That’s a module path and it’s not related to the udev labeling.

Notice that the rule below works just fine.

# Allow access to the Udisks2 API
dbus (receive, send)
    bus=system
    path=/org/freedesktop/UDisks2/**
    interface=org.freedesktop.UDisks2.*
    peer=(label="snap.screenly-client.netconfig"),

I guess - something wrong is with another rule

dbus (send)
    bus=system
    path=/org/freedesktop/UDisks2/**
    interface=org.freedesktop.DBus.Properties
    member=PropertiesChanged  # <==== Maybe here? I don't know what does it mean, unforutnately
    peer=(label="snap.screenly-client.netconfig"),