So you are using a 12 V battery for the square wave generator circuit? And you are using the bench supply for the voltage translation circuit? How are you connecting the circuits to establish a common ground? Is the bench supply bipolar?
Are you planning on using the battery and the bench supply for this system after you have finished developing it?
I have no idea what you mean by "high to LFO". How about some values? You lowest frequency square wave of interest is how many Hz? You highest frequency square wave of interest is how many Hz?
I'm in the studio today in about four hours so I'll post the frequency range in Hz. An LFO is a synth thing it means: Low Frequency Oscillator.Actual numbers...? To repeat post #17:
"What is the minimum and maximum frequency of your square wave that needs to work?"
Also, with a single power supply you cannot get an output signal that extends down to -5 V (5 volts below ground) without some capacitors somewhere. Post #2/4 is the easiest if you can tolerate the startup transient. (BTW, a 555 has one also.) More complicated is building a charge pump to produce something around -10 V for the negative rail of the opamp in post #6. See that V- in your schematic? That ain't ground.
ak