You don't need 300, though it won't hurt. 10 minutes at 250 will do it.
However, if you don't rinse very well first, whatever was in solution, ie. detergent, or dirt that wasn't washed out, will be left after the water is gone.
So, before drying, go outside and holding the chain in the middle spin it like a windmill to drive most of the water and whatever is trapped out. Just as a spin cycle is important to rinse your laundry well, so it is for your chain before you put it in the drier.
BTW - I agree with the others here who said, no water, wash with mineral spirits.
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