I checked that bricks and mortar do not provide enough of a ground path to trip a GFCI, so,
if you have no ground and no GFCI protection for this lamp, IMHO there is another choice.
You can hook up a (make or buy) indicating LED that runs on 120v and pulls 10 mA or so, much like the LEDs that are in some GFCIs.
This red LED is wired between the ungrounded lamp housing and the neutral wire. Then, if the hot lead shorts to the lamp housing, the LED will light which indicates an unsafe condition. The LED is small enough that you can mount it almost anywhere near or on the lamp.
I could not find a downside to this idea. It works on paper but I've never done it.