I realize I might lose some accuracy by going this route, which I think I can mitigate by averaging multiple readings
No .... taking an average only compensates for random noise. What you will have is systematic inaccuracies. These are caused by the DC offset on the op amps.
I do see on the schematic where the max chip uses two op amps
Yes but do you see all those switches on the inputs. They are used to compensate for the DC offset on the op amp's input. If you try to get a lot of gain from an amplifier at DC then you amplify the DC offset as well. This causes the amplifier to saturate without even having any signal from the thermocouple. Any circuit used must compensate for this.
Why don't you try an google a schematic?