250W is way too hot especially in a small area. The reptile bulb should work fine, I swapped in a 60 or 75W reptile bulb (preferably no light, they have ceramic ones that only emit heat, but if no other option one that emits light is ok too) for the red 250W and it was much more tolerable for me and the chicks. If you have a thermometer you want the ambient temperature under the bulb to be roughly at 90-95F if the chick is under a week old.
Other alternative that works very well: if you have a heating pad withOUT an auto shut off, you can wrap that in a towel, pillow case, or Glad Press N Seal, and let the chick press against that for heat. You won't need to check the ambient temperature in a set up like that.