You are making assumptions about how the monitor power supply works without any basis for them. A 12V adapter is unlikely to damage the components, but it is possible depending on the design and that's why I mentioned it. The likelihood is probably vey tiny though so you can probably ignore that bit. The rise in current you mention would happen on the input to the monitor and therefore it would have to be able to handle it.
The simplest answer if you have one is plug it in and see, I just can't 100% guarantee it's safe to do so (probably is)
Why would you assume the monitor uses 12V internally, and that it steps it down efficiently. Most logic circuits are less than 5V. Most likely the 19V input is converted to several different voltages for various circuits within the screen maybe with DC-DC circuitry but possibly just using simple regulators for some of it (which just waste the excess voltage).
With a regulator providing your input voltage is high enough to provide the output voltage it will work just fine, with a DC-DC converter if the input voltage lowers the current will have to rise to compensate. Either will work over a range of voltages but you can't just assume 12V is enough, if it was why wouldn't they supply a 12V supply to begin with.