Could AXE132 drive this LCD?
Very probably, but it's worth looking at the AXE134Y 20x4 OLED which seems comparably priced to what you have. It will be very much easier to interface to using serial.
Just for amusement; to send "Hello World" to this I2C display would require something like -
HI2cOut $14, ( $34, $14, $18, $38, $18, $16, $36, $16, $15, $35, $15, $16, $36, $16, $1C, $3C, $1C, $16, $36, $16, $1C, $3C, $1C, $16, $36, $16, $1F, $3F, $1F, $17, $37, $17, $17, $37, $17, $12, $32, $12, $10, $30, $10, $15, $35, $15, $17, $37, $17, $16, $36, $16, $1F, $3F, $1F, $17, $37, $17, $12, $32, $12, $16, $36, $16, $1C, $3C, $1C, $16, $36, $16, $14, $34, $14 )
That's a full 113 bytes or so compared to the 24 used for serial output, and you have to work out what each byte to send is.
That's obviously not the way to have to do it. Applying hardware abstraction would allow just the output routines to be written similar to below. The SYMBOL commands have to be set right and you mustn't use 'b1' elsewhere in the code. Every byte sent to the display has to be via a call to one of the two routines. That's far less efficient or as easy to use as serial.
Code:
Symbol bitD0 = bit8
Symbol bitD1 = bit9
Symbol bitD2 = bit10
Symbol bitD3 = bit11
Symbol bitRS = bit12
Symbol bitE = bit13
SendB0AsCommandByte:
bitRS = 0
SendB0AsDataByte:
bitD0 = bit4 ; Send msb first
bitD1 = bit5
bitD2 = bit6
bitD3 = bit7
bitE = 1
b2 = b1 ; b2 holds msb with E set
bitE = 0 ; b1 holds msb with E clear
HI2cOut b1, ( b2, b1 )
bitD0 = bit0 ; Send lsb second
bitD1 = bit1
bitD2 = bit2
bitD3 = bit3
bitE = 1
b2 = b1 ; b2 holds lsb with E set
bitE = 0 ; b1 holds lsb with E clear
HI2cOut b1, ( b2, b1 )
bitRS = 1 ; Send data byte next time
Return