> if it dies
It won't die.
Reverse breakdown is not fatal if the power rating is not exceeded, if the current is limited. And you have the resistor.
Say the LED reverse breakdown really is 5V. (It is probably more like 7V; 5V is a CYA spec.)
The 5Vrms winding peaks at 7V.
The excess, 7V-5V = 2V, appears across the 5K resistor.
2V/5K = 0.4mA.
P = V*I. The LED feels 5V at 0.4mA which is 2mW.
What is the power rating of an LED?
We "know" that any little semiconductor is good for 50mW. Mostly these-days, 300mW. There is a spec that the forward power can be 1.7V*20mA or 34mW.
The 2mW of the reverse breakdown is far-far below this.
You "can" get in trouble at high forward current and much higher supply voltage. Hmmmm.... lets try forward-bias half-wave 40mA (to get 20mA average) and a 120V supply. Use a 3K resistor. The reverse-breakdown is 5V at 38mA, 191mW. ~~200mW is "probably" safe, but nothing on the data-sheet assures us of that. (Also note that this condition puts 5 Watts in the resistor-- getting 1.7V from high volts is very inefficient.)
If you want to be truly paranoid, put a 1N4007 backward across the LED. Now the reverse voltage won't be over 0.6V.
Filtering *may* be nice. The half-wave action gives a 60Hz flicker, worse than the 120V flicker of a Neon or Fluorescent. If you move your head fast you can see it. If your youth involved mind-altering drugs, it may bring-back odd illusions. I didn't mind giving a half-wave LED indicator to students, but if I had to work around it much longer it wudda got a proper rectifier and a capacitor.