How do I solder a ~20-pin USBc connector near heat-sensitive components?
You cant do this without the right tools for the job. Based on your description, you also need to shunt away heat from sensitive components. Kapton tape might help. I've heard of people using aluminum foil, but thats conductive and you have a battery powered circuit. I also strongly recommend getting a microscope for electronics rework. The pitch of the pins on that connector is so small you will need magnification. 

You could try removing all the solder with solder wick, but thats challenging to do with SMD components. 

You could try using extra solder to heat up the entirety of the connector's pads all at once, but you run the risk of lifting pads at that point.

My personal recommendation is to use a hot air gun and shield off the sensitive components with the tape. If you try to use the wrongs tools, you will irreparably damage your phone.
Also, make sure your phone is entirely backed up before attempting this. I would approach this repair as if the phone was going to be unrecoverably damaged during the rework.

Based on my understanding of your experience level working with sensitive electronics, I would also recommend practicing on something else first. The risk of damaging your phone is too high.