Should I go with a stainless steel or copper based tankless water heater?
Instead of focusing on the materials, you should focus on the significant differences between condensing and non-condensing units. 

Condensing units are more efficient but can be more difficult (read: expensive) to install as a retrofit because they generate acidic condensate (water) and you have to be able to get rid of that water. This is a big deal if they are installed inside the home, maybe less of a big deal with an exterior installation. This being said, there are standard solutions for the problem.

Non-condensing units are less efficient but you do not have the condensate issue mentioned above.

Both types require a different venting system then your traditional tank heater. Condensing units exhaust cool vent gases and can be vented with cheap PVC pipe. Non-condensing units exhaust hot acidic gas and must be vented with special corrosion-resistant vent piping.

Manufacturers require that their units be installed by qualified persons. I would suggest (I know this is a DIY site, but...) that you consult with a qualified installer, as they would be able to outline the difficulties and cost differences between the two types in your specific circumstances.