Is it possible to use the HX711 as an amplifier without using its ADC capabilities?
No, there is no facility (or pin) on that device that gives you the analogue output. Either use the digital output or choose another chip. An idea would be to use a differential bridge amplifier with in-built digitally controlled programmable gain such as the LTC6915: -



There are plenty of devices out there that would appear to fit the bill. The LTC6915 is just an example of one of them. Check offerings from Microchip, Texas Instruments, Analogue devices, Maxim etc..