I now know that my external feedback loop interferes with the internal loop circuitry. In what way can I mod the above circuit so that it will still serve as a summing amplifier for voltages V1 and V2?
That was an expensive mistake. PGA207 is a $30+ part. You can easily get same performance in your application using $3 worth of parts, or less.
Given the relatively low prices of board manufacturing today, I would say you might be losing money by spending any time thinking about using this circuit as-is.
I bet you it'll be overall cheaper for you to tweak the board to use a normal op-amp. You're not even using the adjustable digital gain feature on the PGA207.
PGA207 is a $30+ instrumentation amplifier with digitally adjustable gain, when a comparable resistor-adjustable gain instrumentation amplifier would have cost $5 or less, say AD623, AD8226, AD8221, etc. (while still making the same mistake).
Almost any modern precision op-amp costing about $3 in small volumes will beat PGA207's performance in the system anyway, and you can do even better if the voltages are limited to +/-2.5V - since there's a whole slew of excellent R-R op-amps that work up to 5V.