The esp8266 WiFi module shouldn't be too hard if you just have to shout AT commands at it.
The biggest hurdle will be the available programming space. The shipped firmware is 28,656 bytes of a total 28,672 bytes available on the ATmega32U4. That leaves a mere 16 bytes! However, there are a few options.
Each T400 secretly shipped with a 1MB SPI flash chip to store data. This is well suited for storing the thermocouple lookup table, fonts, strings, and any other static data. This would free up a good deal of space on the 32u4 for further hacking. This is the ideal solution. See https://github.com/PaxInstruments/t400-firmware/issues/191 for background.
You could get some extra space by removing the SD or LCD functionality, which both require some pretty bulky libraries. If you know the temperature range you need, you can truncate the thermocouple lookup table to remove unnecessary values. These are less than ideal, but more expedient solutions. They could work if they meet your needs.
If you open the enclosure, you'll see two unpopulated headers. One if a regular ICSP header and the other is a breakout header for SPI SDA and SCL lines, 3.3V, GND, and two data pins. The ribs inside the enclosure are located specifically such that a hole can be cut to accomodate a 2x3 cable. See https://github.com/PaxInstruments/t400-enclosure/issues/63 and https://github.com/PaxInstruments/t400-enclosure/issues/29 for background.
I was working on a little developer version of the T400 that could possibly fit little modules inside the enclosure. I made a document with a bunch of reference designs and resources if you want to take a look. See https://github.com/PaxInstruments/t400d-modules/blob/master/moduleIdeas/t400d-modules.md