Well, as Bryan (the developer) said, he built that (first) thermocouple board up and didn't have good luck with it, so he abandoned that design and moved on to the circuit is found on the 4.2.X boards. I don't know of anyone else that has used that little board you have there....
I have built the new v4.2 circuit and it works great for me and I haven't heard of it not working for any of the v4.2 beta testers, so I think this is a good circuit and I would recommend that you work with that design. I think the circuit design is fairly similar to what you have there, but some of the component values are changed, and there are a few resistors added at the thermocouple connection, so I think you COULD use that little board as a base to build the new circuit onto if you note the changes and add the missing components. I would abandon the idea of piggy back soldering the board to your main HM board, that is a dead concept, I would wire it in manually. I would also caution you about your terminal block connection because a thermocouple connection is actually quite specialized, the terminals need to be made of certain metals to match you thermocouple type otherwise it will influence the temperature readings. So, you need to use a real thermocouple connector to make that connection. You can get inline thermocouple connectors, and panel mount connectors as well, I would suggest you look into one of those options.
I have done quite a bit of experimentation with the thermocouple at this point. I have used the thermocouple through a long CAT5 cable, wiring a thermocouple connector to the copper wire and running the raw thermocouple output down the CAT5 cable. This "worked" but required quite a bit of offset to read the proper temperature, and the required offset would change as the temperature differential between the thermocouple junction and the thermocouple amp (inside the heater meter) changed, so that's a bad idea. I have also experimented with running an external thermocouple amp out at the grill, with the thermocouple plugged into a proper inline thermocouple connector with a very short bit of copper wire running from that to the (external) amp, then running the output of this amp down the same long CAT5 cable. This works great, requires no offset at normal ambient temps, however I have only been able to test it in ambient temperatures from around 55F to 90F, it did work reliably in that temperature range....
SO.... I would tell you to first get ahold of the new thermocouple amp circuit and mod your board to match that design (I built my external amp on an IC socket using "flying leads" on all of the components, so it can be done and is not THAT picky of a circuit). Then I would tell you to either get a real thermocouple connector and connect it to your amp. You could build the amp into the HM and use a panel mount connector mounted in your HM case (so the amp and connector are held at the same temperature) or you could use the amp outside of the HM using an inline thermocouple connector (the critical thing here is the amp and the thermocouple connector/junction must be held at the same temp). You can connect the thermocouple through the CAT5 cable by running 3.3v and ground out of the HM on two spare CAT5 wires, then running the output of the amp back down a third spare wire, which you would then jumper over to one of your probe leads inside the HM. You must remove the pullup resistor for this probe, and I would recommend that you install the RC filter (resistor and capacitor) inline with the thermocouple to remove noise from the signal. I think I said all this before, but you seem to still have questions so I am trying to help you out here...