It's more likely that your iron is under powered for your application, and frankly a copper tip will likely make the problem even worse. There are three dimensions that impact heat transfer (temperature being equal): Thermal mass of the tip, thermal conductivity of the tip, and power transfer to the tip.

If your iron cannot support a regular tip, it's likely that it cannot get enough power to the tip to sustain the heat being pulled from it (copper will just expedite this). You can try a beefier tip (or load up a tip with wet solder which can kinda give the same effect) or you can get a higher power iron.

I've been soldering for 15 years with all manner of irons, I've never needed a copper tip, but I often need more than your standard hakko or weller. 400C on a the hakko is not at all close to being equal to 400C on a Metcal for instance.