If (!) the lcd uses a segment per pin, you could indeed read it with a 16 bit piso-shiftregister.
A shift register usually uses 3 lines btw.
I wouldn't solder one on the LCD-pads of the PCB yet in case I'm incorrect.
Even if... it uses a pin per segment, you should be sure you're lucky the input lines of the register match the pin-out of the LCD. Soldering the extra wires to power and command the chip also seems quite difficult.
I'd probably try to solder thin wires to each LCD-pad and use an arduino to check further first.
If it works I'd use a big blob of hot glue to keep the wires in place/assure there's no tension and consider the PH-meter a prototype.
If... it uses a pin per segment you could use a shift register, but using an Atmega8 (oldest arduino, $1.20-chip) may be interesting as well.
It has enough pins to read the meter and gives you the possibility to create an intelligent PH-meter readable by rs-232, rs485, I2C, SPI (shiftregister) or other protocol. If it would keep a pin high for PH-value x milliseconds every 15 milliseconds, you could even read it with one pin.