For example, while the original battery capacity for Apple Watch Series 1 (38 mm) is 205 mAh, can I replace it with a 273 mAh (38 mm) battery (originally for Apple Watch Series 2, 38 mm), and assuming if it work, would extend the standby time for Apple Watch Series 1 (38 mm) much longer than the original design? Will it work, and be safe?
Changing any lithium ion battery that isn’t enclosed in protective metal is unsafe. If the part is crimped or pinched or you get a screw loose, the consequence is fire. Just like a home repair of a gas line or working on live electricity isn’t safe in general. It doesn’t mean some people can’t do it, but that most people shouldn’t without some education and training and the willingness to start over if they mess up with one part.
Apple gear is so small and light that the internals can be tricky to repair so I’d take it to someone that does dozens of these swaps a week or more if you concern is about safe.
It turned out that the battery of difference series of Apple Watch is not compatible with each other, due to their connector interfaces are very different. [reference] So, as much as safety has to be concerned, the idea won't work in the first place.