Not all batteries are the same...
Alkaline chemistry starts at a higher voltage, then, as it is used up, the voltage drops off. This drop off is steady. Frequently, as you have experienced, the voltage will sag below what it is supposed to be.
Devices that use batteries usually will operate over a range of voltages, so they continue to work as the voltage sags, but at a reduced output. You can see this with the remote control for your TV. As the batteries are used up, the range that the remote control will work is reduced. If you normally sit in close proximity to the TV you will not notice this reduction, until you go across the room and try to use the remote.
In devices that have to operate at a specific voltage, a voltage regulator is built into the device to hold the voltage constant. This adds to the cost of the device, and is often omitted from the design. I have no idea of the actual output voltage of your device is critical or not. Perhaps Dr. Beck can give you a lower voltage limit.
You have a couple of options.
You can change out Alkaline batteries frequently and only use part of their available capacity. Or you can change to a rechargeable chemistry.
NiMh batteries have lower capacity, but instead of a gradual drop off of voltage over their life, they maintain around 1.2 volts per cell until the very end, then the voltage plummets rapidly. This results in a kind of "voltage regulation" for the device. Instead of the output voltage gradually decreasing, it holds steady until the very end.
Speaking of capacity, an Alkaline 9 volt battery under a 100 mA load has a capacity of around 400 mAh. A NiMh 9 volt battery has a capacity of around 200 - 250 mAh. Unfortunately, with the Alkaline battery, the voltage drops off after you have used about a third of its total capacity. This drops it usable capacity down to around 130 mAh. Under these conditions, the NiMh battery will actually last longer than the Alkaline battery.
There are two versions of 9V NiMh batteries. The most available, and cheapest, is made using 7 cells. Each cell operates at 1.2 volts, so the operating voltage of these batteries is actually 8.4 volts. They other version is made using 8 cells. Its operating voltage is 9.6 volts.
The problem with the 8 cell battery is that it is physically larger and sometimes it won't fit into the battery compartment in the device.
If the 8 cell battery will fit, it will give you about 1.3 volts more than the 7 cell battery, and the 7 cell battery will give you around 0.5 volts more than the Alkaline battery.
The downside is that you will have to charge your battery more often, the upside is that your device operates at the voltage range it is supposed to.
If you purchase based on price, you will end up with junk and will have problems. You need to find quality batteries, then find the best price on them.
Quality Alkaline 9 volt batteries can be found for around $1.50 per battery, if you order a quantity. Junk batteries can be found for as little as $0.50, but their performance can be very bad.
Quality NiMh 9 volt batteries run around $10.00. Now, the NiMh battery is only as good as its charger, so a quality charger would run another $30.00. This gives a total of $40.00 for the whole set up.
You said you are getting about a week from an Alkaline battery, so in order to break even you would have to use your device for a little under 30 weeks. After this, you would be saving money by using the NiMh battery.
There is a Lithium Polymer 9 volt battery available. It is made up of two cells and has a peak voltage of 8.4 volts, but these cells operate for most of their run at 3.7 volts per cell, so you end up with 7.4 volts, which is less than the 7 cell NiMh battery.
You also need a specific charger for charging Lithium Polymer batteries. If you try to charge them on a NiMh charger, they can explode and catch on fire.
These are fairly new, and not a lot of testing has been done on them. They seem to work fine "out of the box," but little is know about how long they will continue to perform.
The bottom line is that rechargeable batteries will work very well, if you start with quality batteries, and use a quality charger. The discussion about the "memory effect" is way overstated, and is easy to avoid. However, rechargeable batteries do not store well. If you sit them on the shelf, they will die and if you try to revive them after a long storage period, they will under perform. Rechargeable batteries make sense if you are using the device daily, and plan to continue to use it frequently.