This isn't really my area, but I believe one solution would be to use a capacitor and buffer. The output of the circuit (or of another buffer) feeds into the cap which then feeds into the buffer. When you want to freeze it, just disconnect the cap from the circuit. The cap continues to feed the buffer which outputs to the display driver. The main downside is that the cap slows the response time, but careful selection of suitable buffer(s) and cap size should make it manageable.
A quick look at the 7106 datasheet mentions a 1pA maximum input current, so really you probably don't even need a buffer after the cap, you probably aren't going to get much higher impedance than that anyway.