The magnetic loop antenna is the answer to the question: "how do I make an antenna much smaller than a wavelength but which will actually radiate some power and not dissipate it all in its own losses?"
The way it does this is by being as high a Q circuit as possible, so that with a small amount of power I can make huge circulating currents and enormous electric fields.
So it seems that any effort to reduce the Q, with losses not provided by radiation resistance, will throw away some of your power. So go ahead, put some extra dissipation in the system (series resistance, capacitor losses, whatever) and you'll increase the bandwidth, but you'll also reduce your radiated power.
One possible avenue would be to make an array of loops each tuned to a slightly different frequency, and fed with some suitable network. Maybe you could get more bandwidth that way. Hmm, you could even arrange them like a log periodic array and get some gain!