The best strategy in such a case is to use a variable regulated power supply and turn up the voltage until the device works as it should.
You don't have such a supply, but you can try higher and higher voltages until it works. You already did some testing like this and it is obvious that you need a bigger power supply, probably 12 Volt, but you may need more to get it to work.
Get a bigger 12 DC output supply and try it. One that can handle the current without melting. Hopefully that will solve the problem. It should have the max current written on it, but the weight of the supply is a good indicator too, of the current capacity, you simply need a bigger, heavier 12 volt DC supply.
If the machine still doesn't work you need to try a slightly higher voltage. The device may be made for 15 volt and at 12 volt it just lits up but doesn't work properly.
You do not know if the device needs well regulated voltage or if it can take raw rectified Voltage. If the device is very valuable you might want to consider buying a regulated supply, and maybe with variable voltage too, but that is expensive. Most devices with external power supply are fairly tolerant when it comes to voltage. A few volts too much is no problem. But, on the other hand, your device may be an exception.