The current rating of the power supply is unimportant - sort of - as long as it is at least enough. I say "sort of", because it depends on whether or not the supply is regulated. If it is, then it doesn't matter how little, or how much, up to the rated current level, that you draw from it. The voltage will remain at 9v give or take. If the supply is unregulated, then the chances are that it will be rated at 9v nominal, only when it's loaded close to its maximum. At lower currents, the voltage could well be 20% above that, particularly with small transformers which are notorious for poor load regulation. So the output could easily be up around 11v when lightly loaded.
Your bark trainer might well tolerate this, even though it represents getting on for 100% over-voltage, but it is a risk that I don't think I would want to take for the sake of shelling out the small amount that a suitable 6v regulated wall wart would cost.
ElectronDepot website is not affiliated with any of the manufacturers or service providers discussed here.
All logos and trade names are the property of their respective owners.