How can I solder this circuit board without breaking it?
I don't really see you'll have a problem, for the resistor to move the solder on both ends would have to be molten. For that to happen a lot of heat would have to conduct through the resistor and at that stage it'd be getting it so hot that damaging the PCB and/or resistor would be likely.

I'd just start by soldering one side as per usual and then give it a minute or two until it's cooled entirely before starting on the other side.