Westley – That’s definitely not a Dominant E… Looks like it could already be a Pirastro Gold Label with the color of the sleeve on top of the bridge. What color is the winding at the peg end? Those strings are definitely bright, but I wouldn’t necessarily call them “shrill” – I suppose it depends on your sound preference.
As for the bow – Yes, a heavier bow will always help you control bounce. But that’s not necessarily what you want all the time. Relaxed bow hand and arm, and good bowing technique should smooth out unwanted bouncing just as much as the weight. If a bow is too heavy you won’t be able to accomplish more of the advanced technique (quick spiccato, ricochet, etc). Make your decision with this in mind.
When I’m choosing a new bow, my primary concern is how well it’s able to pull out a full tone while still being light enough to move quickly. This is the balance you’re looking for. If the Pietro is able to provide a rich and full tone, you *will* want it to be a lighter stick as you progress. If it’s not able to maintain grip, or you don’t like the quality of tone it provides, move on to another bow and find a better match.