The solution is not to use the wall switch to control the fan, but the remote, as the product is designed; and to produce a wall holder for the remote which is placed such that people coming into the room are guided to use the remote rather than the switch, and such that they can invoke the "light on" function of the remote without removing from the holder, so that the remote acts as the de facto wall-mounted light switch for the room.
The issue is clearly that when you walk into the room, the main switch is always in its expected location, which reinforces the behavior of reaching for it, whereas you have to look for the remote. If the remote is in a holder that is more conveniently reachable than the switch and as easy to operate, that problem will solve itself via user behavior modification.
You could also make modifications to the wall switch to make it less convenient to operate. For instance, if it is a large, flat rocker switch, build a cover for it so that the rocker can only be operated through finger-sized holes in the cover, discouraging regular use.
If these Hunter Fan people were more clever, they would sell or include junction box covers that double as remote control holders. That way users could uninstall their light switch (wiring the circuit so that it is permanently on) and replace the switch panel with the remote control holder panel, so that the remote control now appears exactly where the switch used to be.
"Modified Sine" outputs are very bad approximations of AC
This is a capture of the output of an APC 650 recorded by Jesse Kovach, while under load.
Notice the severe over-amplitude events at the extremes (the spikes at top and bottom). In reality they are actually much greater in amplitude, but the oscilloscope in the image was not fast enough to capture it.
Sharp edges in the time domain equate to broad-spectrum noise in the frequency domain. All of this high-frequency content represents additional energy that must be absorbed by protection circuits. If not, it can exceed isolation withstanding limits in the various input stage components and "burn through". If this doesn't burn out the input it will result in a cascading failure where it will cause something to fail on the secondary side from the resulting overvoltage.
...and that's just one failure mode. There are others. Psuedo-sine waves are poor matches to sine-wave inputs. :(
Go DC-DC instead of DC-AC-DC
A much better (and much more efficient!) approach is to go DC-to-DC directly (note: you can't actually go DC-to-DC directly if your input voltage is lower than your output voltage, but the details of this are well contained inside a "DC-DC converter").
Self-contained switch-mode power supplies for Dell laptops that take DC inputs are available in the marketplace. Here's an example:
which I sourced from:
http://www.amazon.com/Adapter-Charger-Dell-Latitude-D630/dp/B002BK7JEC#
Please note that I have no personal experience with this particular product and many cheap DC converters are poorly designed internally. Be careful.
Best Answer
Okay.
You clearly mean ampere hours.
If it uses 4.5 A when being fed 12 V, then you can assume that with 9 V battery that it will drain roughly \$4.5\text{ A}×\frac{9\text{ V}}{12\text{ V}}=3.375\text{A} = 3375\text{ mA}\$.
Let's assume you will run it for 15 minutes, then you will need a 9 V battery with at least \$3375\text{ mA}×\frac{15\text{ min}}{60\text{ min}}=843.75\text{ mAh}\$.
Considering that you will be discharging it at 4 times the capacity, then I'd aim for a little bit higher capacity, say 2000 mAh or even higher, or put several 9V batteries in parallel. Because high discharge (several amperes) will lower the capacity.
If you increase the voltage, then the ampere will also increase.