Will a 1500VA autotransformer do the job for an appliance with 13A and 120V?
You wouldn't starve your device. You might cause the transformer to run a little warmer than normal but probably not by a measurable amount. Although transformer current is also limited by the magnetic field in the core, this limit won't be so close to the rating that you should be concerned. 

The rating plate may not be this clear but I strongly suspect that it's not that your appliance draws 13A, but that it's fused at 13A, drawing less. Initial current may be another matter especially if most of the load is a motor. 

If I already had the transformer I'd try it (but maybe measure the current). If I was buying the transformer I'd probably go for the next size up if possible