| Normal relays/contactors are made to draw as little current as possible, so their coil have high impedance. Unless you actually find a current relay or perhaps rewind one, anything rated for connection to a voltage source (even 12V) have too high impedance for going in series with the lamp
You might look into connecting the relay coil in parallel with the ballast for a choke ballast (may or may not be applicable with autotransformer, depending on which taps are available from outside). Only applicable in circuits with superimposed ignitors or Probe Start. It may add some failure modes to your system tho :
1. AC relays draw higher current momentarily, from the moment when the coil is energized to the moment when the core is pulled in and contacts are closed. (The relay is effectively a choke with big air gap before the core pulls in, and close to zero air gap once it does). The lamp may or may not like this very short time overcurrent. (I assume that once the contactor pulled in, the curret will be low enough to fall within the lamp spec tolerances)
2. If the relay gets mechanically stuck in the "not pulled in" position, it will pass the higher current continuously - Which will lead to its coil burning up. Once the coil shorts, the lamp will likely be destroyed as well. A relay may get into this position if it fails mechanically for whatever reason, or if you supply it's coil with voltage that is not sufficient to pull in - The greatest danger i guess is at voltages around half of the relay's rated voltage
There are current relays that directly sense current, this would be the best solution for current sensing instead of all of the above
The relay with NC contact + staircase lighting timer cover all what you ask for
Or build your own electronic control with a means for current sensing (transformer, resistor, etc), sense amplifier, the logic, and output control with a triac or relay
Alternatively you might build a system that works on time alone, without knowing the real state of the MH lamp : Power on the Halogen for a set time after each power on, reset timer at power off. This is pretty much what a compressor protection module does, except you need the NC contact. If its not provided, add a second relay to it to invert the output
Some compressor protection modules are smarter than that - they dont have the delay at first power on, only at "hot restrike". This makes sense for the lamp use as well, since at first power on the lamp astrikes right away, and at second powe on you have to wait before it restrikes. If you want to have some time delay into the lamp warm up, either get a "stupid" module (but it will power the lamp unneccessary long at first power on), or a "smart" one only for the hot restrike delay + staircase timer for the additional constant time delay
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