Is there a way to test whether the UART Tx buffer is empty without waiting?
It doesn't look like that function is implemented. The source for HardwareSerial.{cpp,h} are in /Applications/Arduino.app/Contents/Resources/Java/hardware/arduino/cores/arduino under OS-X. (Someone better versed in Window than I may have to chime in here to point you to that source location.)

Taking HardwareSerial::available() as a model, create a new function, say, HardwareSerial::txremaining(), that does the same thing, but for the transmit buffer.

Left as an exercise for the student:Make a common function that returns the contents of an arbitrary ring-buffer and use it in both ::available() and ::txremaining().