What I mean by "trying too hard" is that the solution is simple. Purchasing a new light adds unnecessary complexity and will probably do nothing to achieve the goal.
Wrap a piece of paper around the end of the outside of the strobe and tape it in place as a cylinder to make a wide snoot. You can make the cylinder into a cone to vary the aperture of the end of the snoot to create a larger or smaller spot of light. Varying the length of the cone/cylinder, and the distance between the light and the background will vary the resulting light fall off.
It might not be a glamorous solution, but at the least it will give you an idea what is possible, and maybe enough information to know what type/size "proper" snoot to purchase.