It wouldn't be a hard short to ground or the lights would never come on, and the breaker would trip immediately. I'm assuming from the original post that the lights do come on for the 1 to 2 minutes until the breaker trips? If so, then it's not a hard short. Sounds like for whatever reason you're drawing just a little bit more current than you think you are. And there are various possible reasons why.
First of all, a 55 W halogen bulb might very well draw more than the 3.8 A that Tim calculated. This bulb is likely specified for 55 W at 12.8 V, so its current draw would be 55 W / 12.8V = 4.3 A at 12.8V. And assuming the filament acts as a simple resistive load in the steady state (probably a reasonable approximation), it'll draw even more current at a higher voltage produced by your alternator, say 4.3 A x (14.5 V / 12.8 V) = 4.9 A at 14.5 V.
Also, these light bulbs are not built or tested to super tight tolerances, so I wouldn't be surprised if they vary quite a bit from one to the next.
Also, H3 bulbs come in various power ratings. Are you sure you have 55 W bulbs and not, say, 100 W bulbs?
Anyway, the advice given by Tim to isolate and test one side at a time is good advice. And I would add to that that you should also use an ammeter to actually measure the current draw. Then you'll know rather than guess what you're actually dealing with. That always helps debug a problem.
And finally, DO NOT EVER arbitrarily use a higher rated breaker just because the existing one is tripping. Remember that the breaker is there to protect the wiring from excessive current. The breaker must be rated at less than the current carrying capacity of the wire, so that the breaker trips before the wire can heat up enough to melt or to ignite the insulation. If you use a breaker whose rating exceeds the current carrying capacity of the wire, you create a FIRE HAZARD!!!
If it turns out that the normal load current of your landing lights really is just bigger than you thought, and that you really do need a bigger breaker to accommodate that normal load current, then make sure that your choice of wire gauge is also adequate for that current, and is adequately protected by the breaker.