Electronic – the tiniest, low-power, and efficient am transmitter

In general what you are looking for is a analog to digital converter, or A/D for short. There are many different types. Each different technology has its own tradeoff between speed, accuracy, resolution, cost, and other parameters.

One method of making a number from a analog parameter is to have the parameter change the frequency of a oscillator as it seem you did with your 555 timer, then measure the frequency. There are many other ways too. Nowadays, just about every microcontroller comes with a A/D converter built in. 10 bits is quite common. Some low end ones may only have 8 bits, and 12 bits is available too. Beyond that the anlog requirements get difficult to meet with the same technology the micro is fabricated with, so you pretty much need a external A/D for more than 12 bits.

The battery in your UPS may be isolated from the output or not at the same ground reference, so some care must be taken to measure its voltage and get the data into your PC. My first knee jerk reaction is to put a small micro on the battery circuit. It measures the battery voltage locally, then sends the resulting digital data serially over a single opto-coupler to another micro that interfaces to the PC via USB, or perhaps directly to the PC's COM port.

Is the duty cycle supposed to switch from 50% to 0%, or decay from 50% to 0%? Is there any particular precision required on the 50%? Is there any requirement regarding quiescent current?

I would suggest that you start by using a 555 or similar chip to generate a symmetrical sawtooth-ish (doesn't have to have straight sides) wave, and then feed that into one side of a comparator. The other side of the comparator should be connected to a circuit which will generate a voltage that will start at half-rail when the button is first pushed, but will either rise toward VDD or fall toward ground after that. If you can use a SPDT pushbutton, a simple realization of such a circuit would be to connect the center pole of the button to a cap whose other leg is ground. Connect the normally-closed contact to VDD, and the normally-open contact to the comparator input. Connect a resistor from that comparator input to ground.

Conceptually, it would be nice if a single active device could be used to provide the output, without needing the separate comparator. It's possible for the 555 to provide an output whose duty cycle will be affected by an input voltage. Unfortunately, the frequency will also be affected by the input voltage; I'm not sure of any easy way to make a compensated circuit such that the decrease in high time will be balanced by an increase in the low time so as to yield a reasonably-constant frequency (as opposed to having a device generate a 1Khz reference signal which is then shaped by a second active device). A two-chip circuit doesn't seem unreasonable, though.

Here's a simple circuit which will illustrate the concept. For simulation purposes, it uses an op amp, but a comparator would work just as well. One slight note: many comparators ground the output when the + input is higher than the - input, so the polarity would be opposite that of an op amp.