Can I "sum" internal ADC channels to achieve extra bits resolution?
If you dither the signal and average more than 256 samples you can get some apparent resolution (not accuracy) increase, but frankly you’d be far better off ignoring the on-chip ADC and using an external ADC designed for load cell applications.

The one you mention (HX711, made by Avia in Xiamen, China) is only about 40 cents (24-bit delta-sigma with on-board PGA and bandgap reference) so it's not like it's going to break the bank!