Sadly I don't have a fluke MM so I can't give advise on how to implement it, but I have done something similar with my ut-61e+ using an ESP32-S3
BTO gave some solid advice in regards to the ESP32 over an arduino. You don't have to give up the arduino framework (or even the IDE though I could go on a 50 page rant about it, I'm a vscode+pio fanboy) so most boilerplate code is infact compatible with an ESP32
I recommend an ESP32-C6 or C3. Both run on the risk-v architecture which has insanely low power consumption, both are also single core which is better for battery (unless you require a second core for something other than your data logging function?

). The C6 also has BLE 5.3 (C3 has BLE5) which reduces power consumption a tiny bit more, but it's negligible, both are good choices.
In terms of processing power, all three are overkill TBH, it's all about choosing the lowest power consumption imho when it comes to datalogging in 2024
Oh, look to use BLE instead of wifi. Wifi is great if you want to view your live data on your phone from abroad, but you'll likely be next to the multimeter when you're logging, right? If you do want "wireless viewing", BLE is far more power efficient than wifi for waking/sleeping/even idle consumption. What I'd do is have a master ESP32 plugged in with wifi enabled & put a slave in the the fluke 117, send the data from the fluke to the master esp32 and let that one do the heavy lifting.
I have to go eat but I wanted to post the power consumption comparisons / calculate how many hours out of the MM battery when logging you should get. I will prolly do it later
good luck with the project!
oh ps. I just thought, go on github/gitlab and search for your MM's name.
somebody may have done it before, just not documented it on a blog/yt video so google doesn't come up with it. at the very least you can search for the ut61e, which has a bunch of datalogging mods based on the esp32. it should save a lot of programming time as the code should work on the fluke/be fairly universal once you've figured out the io & changed the gpio pins if they use the arduino framework
edit: just remembered, the C6 has zigbee & thread(?) radios. not sure of the bandwidth, if it'd be enough, nor do I know if theyre better for battery... that's something to research. may be better protocols than even ble. speaking of BW, ble averages at 90KBps afair, which is far more than enough for your average MM datalogging 2-6/sec sample rate.