I would suggest finding a small toroidal transformer with a 9 or 12 volt winding. Wind about 5 turns on it with wire suitable for mains voltage and the 4 amps of your heater. These 5 turns will be connected in series with the heater. Insulate the ends of the primary winding as these will not be used but they will have quite a high voltage between them.) Connect a resistor of about 18 ohms across the 9 or 12 volt winding. Use about a 5 watt rated resistor. with the 5 turns connected in series with the heater measure the voltage across the resistor. You are aiming to get about 5 volts across this resistor. You will probably have to adjust this resiator value and / or change the number of turns that you wound on the transformer. When you get a voltage between 4 and 8 volts connect the input of a small bridge rectifier across the resistor and connect an LED in series with about a 220 ohm resistor to the output of the bridge. An alternative method is to have two groups of four diodes (At least 5 amp rating.) in series connected in parallel but with the opposite polarity. Each group of 4 diodes will conduct in one direction so with the forward voltage drops you will have about a 2,8 volts square wave between the ends. Connect two LEDs in parallel but opposite polarity. Connect these via a current limiting resistor (About 56 ohms.) to the ends of the groups of diodes. (An alternative to using 8 individual diodes would be to use two bridge rectifiers (At least 5 amp rating.) On each bridge rectifier short the + and - terminals together then connect the AC inputs to the bridge rectifiers in series. This may be easier than mounting 8 diodes.)
Les.
I made a current transformer for a heater to monitor when the thermostat cut in.
I used an old conventional (non-toroidal) transformer, and I cut away and removed the secondary. I put a bridge rectifier and LED across the primary, and a single turn of the heater supply wire (live only) through the space where the secondary had been. The insulation was still on the single turn of wire. The LED came on when the heater took current.
I estimated that I had around 2000 turns on what had been the primary. The mains is 230 V here, and it was a 3 kW heater taking around 13 A, so that gave me around 6 mA though the LED
My suggestion, if using a toroidal transformer, is to wire to the primary to a bridge rectifier and an LED, or two back-to-back LEDs in parallel. A single pass of the 4 A wire though the middle of the transformer should be enough.
You don't need to worry about the voltages. A small toroidal transformer will have around 500 - 1000 turns on a 120 V primary. When you pass the heater supply wire through the middle, that is one turn, and with the primary shorted, the current will be proportional to the heater current, but reduced by the turns ratio. So a 4 A input, with 500 turns on the output (that was the primary) will give about 8 mA on the secondary. If there are more or less turns on the output (that was the primary) you will get a smaller or larger current.
Do make sure that you have a load connected to the output that can take current in both directions.
A transformer and LED arrangement like that will not cause any noticeable voltage drop, and nothing will get hot.
I don't think that the arrangement using the 6 or 9 V winding, (that was the secondary) will work as well. There are far fewer turns on the 6 or 9 V winding, so you will get more current than you want for an LED. There may also be a problem that you need a bigger magnetic field to get enough voltage for the LED, which would mean that you need more turns of the heater wire, so you could end up with far more current than you want in the 6 or 9 V winding, and then a big resistor to bypass or limit the current.