When you say the generator can be wired fro 240V or 120V, that can mean two different things, and I think the difference is the source of your problem.
A typical 240V power source (like a generator) set up for North America is wired in what's know as a split phase 120V/240V configuration. It is two 120V circuits stacked together to make 240V. It's actually a 240V center tapped winding or transformer if you want to be specific. This is most likely how your generator is set up. In this case, you can pull 240V @27A. Or you can pull a total of 120V @54A, but (and it's an important but), you needs to be equally balanced across the two 120V circuits, i.e. neither can exceed 27A. If this is confusing you, that's because it is indeed confusing until you understand it, and most people don't completely understand it. After all, why should they? If you want to know more, google about AC power systems, and in particular north american power systems. This arrangement is unique to NA and the areas that have copied us.
The other way the generator might be wired up, but it would be unusual, is for the two 120V circuits to be wired in parallel rather than stacked in series. Many generators can be wired up this way, but typically are not, and you need to really know what you are doing to make the change, if it's even possible. In such a configuration, instead of having two 120V, 27A circuits, it has one 120V, 54A circuit and no 240V capability. The sad thing is that for smaller generators like yours, and assuming you have no need for 240V service, having a single 120V circuit is much preferable because you can run these higher loads. But almost nobody sets them up that way. I have an old 4KW Onan RV generator that is set up this way and will produce 33A at 120V. I also rewired a 5KW portable cheapie generator like this a long time ago so it could more easily run larger power tools.