Design pressure would change based on the change in allowable stress as your temperature changed. However, over the range of temperatures (about 160 - 366F) you're considering, and taking into account the material you're dealing with (304L SS), I would expect the change in allowable stress to be nil.
With saturated steam at 160 psig or less, its temperature will be <=366F. You must provide tube side relief because you could block in a water filled tube side and then heat it with shell side steam. Size the PSV by considering how much heat will be transfered and use the thermal expansion of liquid water to figure out the volumetric flowrate to relieve. As long as you do not have superheated steam, you should not generate vapor (steam) on the tube side. (But, there may be other relieving scenarios such as fire where steam could be generated.)
You need to examine all possible overpressure scenarios to decide if you need shell side relief. Right now, I don't see a relief situation, but I have only a few facts. If the shell side could contain liquid, then there would likely be a fire case.
HTH,
Doug