Can I let my bread dough rise in the mixer bowl?
Your metal bowl sitting in your 70°F room is 70°F (at least, if its been sitting there for a bit). Your plastic bowl, or glass bowl, or ceramic bowl, or any other bowl sitting in the same room is also 70°F. They're all actually the same temperature.

Now, given, when you touch the metal bowl, it feels cooler than the plastic one. This is because your finger isn't 70°F, and your body heat is transferred away quicker by metal than (say) plastic.

If the dough is room temperature, it won't matter—heat isn't flowing out of it. The dough itself doesn't generate much heat. If your dough started above room temperature, it will cool a little quicker in a metal bowl. But, if you wanted to keep it from cooling, putting it in a warm place is much more effective.

A stainless steel bowl is fine. I'd avoid aluminum (and copper, if anyone makes a mixer with such a thing) due to them being reactive, especially if you're making sourdough.