I doubt a toroid would help here: the self-capacitance to the core, and the slow-wave resonance through the core, likely give a resonance near 10MHz, at a fairly low impedance (couple kohms?). A core isn't useless though; a ferrite rod would be okay, and a high-mu* core will be effective even with the winding spaced away from it (i.e. use a bobbin with ID > core OD). The effective mu will be small (2-4?) but that's still an advantage over an air core coil.
*"High mu" for a rod core, is at least some hundreds; basically any random grade of ferrite will do.
You can build bobbins out of cardboard if nothing else (but, preferably some actual engineering material like phenolic, fiberglass (stripped PCB if nothing else?), etc.), soaking it in varnish or melted wax for stability and to hold the windings in place. Many hardware store glues are also effective. Bobbins can be stacked on the ferrite rod to make something like a pie-wound choke.
But again, this is in the realm of custom designs, and I'd have to try several arrangements and characterize them to land on the ideal design.
Tim