I have a spool of smaller 3mm rope that has its end attached to a ball. My intention is to use this smaller line as a throwline that gets thrown over a tree branch which is then used to drag a climbing rope over the branch to anchor it to the branch. Since the smaller rope is conveniently spooled to a rope holder, I'm looking for a knot that I can use to secure the middle of the line (not the end) to the climbing rope. If anyone knows of such a knot, the help would be very appreciated.
You might want to look into the "racking bend" which can be used for this purpose. https://www.youtube.com/watch?v=-lu0k1aseUI
You would make a bend on the climbing rope, either by folding the working end onto itself, or by making a bight anywhere on the climbing rope.
Other options would include the icicle hitch or sailor gripping hitch, but I think that those hitches might come undone over time. If you just want a temporary solution, they might work for you.
I'm still unclear with the thin line you're describing, though. You have a heavy ball one end, and a spool on the other? what's preventing you from using the ball as a working end?