I'll offer what help I can with my limited experience. I just built a circuit for a phone handset (carbon mic). I sort of based it off the circuit from the Shure 104 manual website.
http://www.shure.com/idc/groups/tech_pubs/@global_managed/documents/webcontent/us_pro_104c_ug.pdf
The instructions pdf you posted gave a lot of info. 3-5 mA for close talk and 10-18 mA for normal use, and it has a resistance of 200 ohms and they suggest a 1.5V battery. For the Shure 104 it can operate fine at 50mA and can even handle 100mA, and they give a list of resistors to use for different voltages from 4.5V to 24V. My point is that the carbon mic can handle a wide range and it's really going to be up to you to decide what current to use.
I tested with current from about .05mA to 50+mA and I did not hear much difference in sound quality, although the carbon mic in the phone is fairly limited in response anyway. The amount of current really just determines level or volume. What I wound up doing was using a 9V and putting a 20K pot in with a 150 ohm resistor so I can vary the current to give the appropriate level to whatever device I'm going into. I guess that would be like the 100K pot in the schematic you posted. At 150 ohms I was at about 45mA and at 20K I was somewhere around .1mA (I can't remember exactly). Or, another way to put it, at 150 ohms I can go into my line input with a moderate gain setting and at 20K I can go into my mic input at a moderate gain setting. For the audio signal I connected from the carbon mic terminals to a transformer with a cap to block dc. and then the secondary connects to pin 2 and 3. Pin 1 is not connected.
So, it looks like your mic is a double button mic. I believe the unedited schematic you posted is for an a single button mic and has an unbalanced signal out, and the negative of the 1.5V is intended to go to ground which would create a circuit to provide power. Your mic has 3 wires, 1 from each button and ground (+, +, -).
Check out the diagram C of the dual button mic here. This is how the instruction pdf describes it, too.
http://www.tpub.com/content/armycomsystems/ss06036/ss060360023.htm
It looks like you are trying to adapt the schematic for the 3 wires. Both buttons need power and the instruction pdf says that the current to both buttons should be equal. In your edited schematic, you would be providing power to only one button.
Although my mic has only one button, here's the way I thought of it. Draw a figure 8 and put the carbon button at the intersection. So basically you have 2 loops each connecting to the + and - of the carbon button. Make one loop your power and the other loop your audio. So, for one loop, add a battery and resistor (or pot) to limit current. That's all you need for power. You don't want the current limiting resistor for the power to effect the audio signal, that's why I think of a separate loop. For the audio you just need a cap to block dc and a transformer. Connect the secondary of the transformer to pin 2 and pin 3. if you need a volume control, you can connect it after the transformer on the secondary. (That is also stated in your instruction pdf.) I have not tried it without the transformer, but I suppose you could give it a try and just hook it up the same way without the transformer and just use the cap to block dc.
A note about cap values. Recommended values I found were from .1uF to 20uF. But if too low then you can lose low end. So, better to stay up around 10-20uF. I chose 10uF because that's what I had.
So, how would you do it. You might try thinking about it in a similar way. What does it need for power. Take care of that first. Then where does the audio need to come from? When it comes to 2 buttons without using a transformer, I'm not sure. Perhaps someone else will have better info. Would you connect the two wires from the buttons to the + terminal and the ground to the - terminal of the battery and then get the audio from the + and - as well? Or, would you maybe hook up 2 batteries, 1 for each button, with the 2 button wires going to separate + terminals and the ground going to both - terminals of the batteries and the audio coming from the 2 + wires?
I hope it helps. Sorry I can't offer more.
