As I've said multiple times, the hd44780 library normally uses Wire but is also compatible with SoftwareWire.
You have said that you needed to free up analog pins which is why you were looking at SoftwareWire.
How does using SoftwarewareWire and LiquidCrystal_Software_I2C provide a complete solution for you? (it doesn't free up any pins, and uses 2 more digital pins)
The Adafruit_INA219 library will still be using the AVR wire library so if you are trying to free up A4 and A5 because the Arduino board you have uses those analog pins for the h/w i2c pins, then those two pins will still be used by the Adafruit_INA219 library even if you use the LiquidCrystal_Software_I2C library and configure SoftwareWire to use other pins since the pins used by Wire cannot be reconfigured.
i.e. if you use SoftwareWire to run your LCD (regardless of which lcd library you use), and reconfigure the SoftwareWire i2c pins used, the Adafruit_INA219 library will still the same h/w i2c pins used by the Wire library as those pins cannot be reconfigured.
At this point, I am not understanding your actual issue and can not recommend any solutions as I think this appears to be an XY problem.
--- bill