Problem with output swing in LM358
The current driving the LED has to come through R2 (1 kohm). That's fed by the LM358 from its power pin, which means it has to come through R3 (4.7 kohm).  That's part of the problem. 5.7 kohm in series with a LED would probably still light it OK, but I suspect that when the LED is on the 358 browns out and stops working until the supply voltage recovers. If you really want R3 in there, try 4.7 ohm instead of 4.7 kohm.