Andy, this is EXACTLY what I created IndyFont for! A lot of my typesetting is in the linguistics field, and there are not that many professional fonts that sport phonetic characters — or even loose diacritics, to manually position.
You can easily find out the built-in kerning of a font by setting the Kerning to Metrics (which will use the font’s own values), typing the two base characters you want to inspect, and then moving the cursor between those two characters. The value in the Kerning field then indicates the amount! This is between parentheses, such as “(-14)”, because it is an automatic value, just as the value of Leading is shown in parentheses when you use the automatic value.
So if you created a ‘?’ (a t with dot below) and want to know how much kerning is applied in the base font between ‘o’ and ‘t’, simply type them and check the value, and then you can use this same value for the underdot character. This will work for almost all combinations, as long as the underdot itself does not interfere with the character before. I don’t think there are many characters that extend at the bottom, but a combination such as ‘?j’ may need a manually determined value.
There is no easy way to read *all* kerning values out of a font. In modern OpenType fonts there is a thing called “Kerning Class”, which is an entire set of left-hand characters vs. right-hand characters. That means there may be hundreds of possible combinations … (which is why it’s stored as a class in the first place, instead of separate combinations!). If you can find an official .AFM file for your font, this usually lists kerning values for certain common pairs.