Need advice re chance of damage to a stainless steel tray in a very hot oven

The temperature of boiling water is 100° Celsius or 212° Fahrenheit. This is considerably colder than the self-cleaning cycle in your oven. It would be perfectly sufficient to set the oven temperature to about 120°C/240°F to heat the tray. It would not take long to reach the desired temperature for the tray at those lower temperatures since metals conduct heat well.

If it is non-magnetic (304) it could be sensitized at about 1100 F (very dim red heat) which would reduce corrosion resistance to boiling brine. If it is magnetic (410), I can't imagine you could hurt it in the oven or surface burners. It could ruin bakelite handles. So there is no way you can damage any stainless in a home oven .