Yes, you can perform that ambient isotopy: any oriented embedding $i: B^n \to M^n$ is isotopic to any other. (This is a lemma proven independently by Cerf and Palais1, but the idea is quite clear: shrink the image of $i$ until it's contained in the chart, then take the limit that defines the derivative of a map.)
In particular, if $h$ is your diffeomorphism $B \cup_{\partial B} B^4 \to S^4$, you may isotope the embedding of $h(B^4)$ so that it is the standard inclusion of the north hemisphere. Then $h$ restricts to a diffeomorphism $h: B \to B^4$, where this $B^4$ is the southern hemisphere of $S^4$.
1The references given on this Manifold Atlas page are Palais, Theorem 5.5 (the essential content is Lemma 5.2) and somewhere in Cerf's treatise on embedding spaces.