Upperbound for max degree of k-tree completion
It is known that a graph of treewidth $k$ and maximum degree $\Delta$ has tree partition width at most $O(k\Delta)$. See Wood, arXiv:math/0602507.
From a tree partition of width $O(k\Delta)$ a triangulation of width $O(k\Delta)$ and maximum degree $O(k\Delta^2)$ follows pretty directly (start with the tree partition, make every bag into a clique, make every vertex $v$ in $U$ adjacent to all vertices in all bags appearing together with neighbors of $v$).
So you get a (minimal) triangulation with the bounds above.
Note: I would not be surprised if you can get a better degree bound.. the additional $\Delta$ term feels unnecessary.