Binomial heap multiplying nodes
I think in this scenario you can just bubble-up one of the updated element to restore the max-heap property.

Notice that by the multiplication you have increased the keys in $V$ children. So the trees rooted at the children still have the max-heap property, but maybe $V$ does not anymore. You can repair this, by swapping the largest key of the children with the key of $V$. This restores the max-heap property for the subtree rooted at $V$. You continue by comparing the key of $V$ with the key of the father of $V$ and swap the key is necessary. You do this, until nothing was swapped, or you reach the root - its the typical heapify routine.

Since binomial heaps have depth at most $\log n$ and the degree is also bounded by $\log n$ the whole procedure needs $O(\log n)$.