Are [ɚ] and [ər] interchangeable for (儿化) er-hua?
They should be the same. It is just differences in notation. [ər] is commonly used by Sinologists, while [ɚ] is actual IPA.

In the Beijing dialect (and standard Mandarin), 包儿 should be pronounced as something like [paʊɻ] (probably written by Sinologists as [paur]). It looks like you're looking at something like Sichuanese, where 包儿 is pronounced as [pɚ].

Take a look at the Wikipedia article on erhua for more details.