I would use silicone rubber cable for the internal wiring and also be careful not to change any of the specs of the fryer as then you will be the "manufacturer".
Depends on the temp' and environmental exposure of the cable, what is the existing wired in?
Seen the glass fibre coated single often used in these conditions.
I would also consider that doing this work will mean you transfer all responsibility for the design and operation of the cooker from the manufacturer to yourself, I would check you are covered by your insurance policy, I would not be taking on such work with a standard insurance policy unless you can confirm you are covered.
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