Is it safe to use a soldering gun when attempting solder repairs to a PCB with IC chips and other processing units?
I would not use soldering gun to solder onto any PCB (it simply does not control the temperature, fast variations/not good). That is not the purpose of the gun. There are better tools for soldering, like soldering iron. I fear that the soldering gun can actually do more damage to the mainboard than the soldering iron. 

I replaced capacitors on my PC mainboard using soldering iron @ 300 degrees Celsius.
To make your soldering succesfull, apply a little solder onto the iron tip, then quickly place it on the solder place for CR2032 (!!! battery must be removed!!!). Add more solder to finish the soldering proccess.

I don't think that soldering the ic for less than 10s can damage the IC (rare), but you should try to be quick as possible. Soldering other components, like battery holder, would not damage other components.

You could apply some soldering flux to the soldering points of the holder. Use quality soldering iron, be as quick as possible and good luck.