All in all, I feel like I have succesfully diversified my multimeter assortment. I have a Fluke 17B I can use for daily work so I can enjoy quick continuity testing and non-irritating power-off features, and I have the 15B+ as a very capable second multimeter, with WiFi connectivity to show trends and to allow for any device with a browser to act as a remote display. It's a shame I couldn't figure out more of the way the main multimeter chip works to get direct access to the display, but the PWM-trick I did to show the IP-address is an alternative that is workable. Here's a demonstration. As you can see, the multimeter has the IP of 192.168.5.117, and that's what I enter into the browser.
As usual, all the code is available for download by doing a git clone on http://git.spritesserver.nl/fluke15bpluswifi.git/. Because esphttpd already was licensed as such, the rest of the code is also licensed under the Beer-Ware license. If you manage to do anything interesting with it, as usual, I'd appreciate a message, I always like to read about what interesting ideas others have. If you want to do anything yourself with the serial protocol, I'd also advise you to take a look in the source code: the fluke15bplus.c gives a fairly thorough description of the protocol details. One last word of warning: if you send something over the calibration port that messes up your multimeters calibration, I'm not responsible. Please do not make any life-or-death decisions with a multimeter that's been messed with, but otherwise: have fun!