Are there downsides to using resistors with too high power ratings?
The only usual downsides that come to mind are size, cost and sometimes resistors with a higher power rating use technologies such as wire wound that can have higher tolerance rating than some others. Other factors such as inductance that may be an issue with RF circuits won't be an issue for a current shunt.

So in general if you're happy with the size, cost and tolerance of the resistors you've found then there's no reason not to use a resistor with a higher power rating than you require.