neu Is it possible to skirt the requirement of going online for the device to check whether the IMEI is carrier-locked, and to unlock the bootloader without doing that?
Many things are possible, with enough money, effort, luck, etc. But smart people worked hard to make this thing extremely difficult, unlikely, expensive, etc.
neu Or, should I just contact my carrier and pay in-full to get them to unlock it?
That depends on your carrier. Some (Verizon) just won't.