Amazon sells a bung with a hole for the airlock and built in thermowell that fits the anvil. It's like $12. I use that and an inkbird 308 to control my hear and cooling. Works perfect. Drop the probe of the inkbird into the thermowell and plug the fridge and heater into the inkbird. https://www.amazon.com/dp/B01KP4D0K2?ref=ppx_pop_mob_ap_share
Is the Inkbird monitoring ambient temp in the fermentation chamber? If so, I would attach the probe to the side of the fermenter covered by some sort of insulation between the probe and ambient, keeping the probe at nearly the internal fermenter temp. Or you can buy a bung with thermowell.
As many have said, I use the Inkbird ITC-308. What I haven't seen mentioned is another cheap addition to this that will improve temp control even more: A thermowell for the temp probe. They're less than $15. So you could get an Inkbird and Thermowel for less than the controller you linked, and it'll be more accurate.
Here is the one I got. it's much better than taping the probe to the side, but not without it's own issues apparently. I feel like my temp readings are more accurate, but can't verify that.
You could carefully cut a bigger hole in the bucket lid, then install a bung with two holes. The airlock goes in one hole and the thermowell goes in the other.
They make them for use in glass carboys. Here’s the one I have. Be sure to measure the thermowell length to see if it will work on your set up (go into the beer) and that the interior diameter works with your probe.
I don’t know this seller but I see this too on Amazon.
Edit: another idea is to keep the existing airlock hole and airlock but also cut a bung hole and add a bung with one hole into which you insert a thermowell.
I wouldn't drill the lid for a thermowell and ball lock port. There are a combo action of items on Amazon you could get and utilize the bung hole that already exists.
Homebrew #7 Stopper Thermowell
Stainless Carbonation Cap Counter
Drill a 1/4" hole in the undrilled stopper for the carb cap and you have a ball lock connector.
The bung with the thermowell works great on my anvil fermenters.
These are pretty inexpensive and the Inkbird probe fits perfectly inside: https://www.amazon.com/dp/B01KP4D0K2/ref=cm_sw_r_cp_api_i_1ErxFbQ81AWZ9
Buy a thermowell, solved all my problems: https://www.amazon.com/Homebrew-Stopper-Thermowell-Weekend-Brewer/dp/B01KP4D0K2
They make bungs with thermowells you can put the probe into, like this https://www.amazon.com/gp/product/B01KP4D0K2/ref=oh_aui_detailpage_o03_s00?ie=UTF8&psc=1
This one works great for me with my itc308 and NB BM bubbler. https://www.amazon.com/Weekend-Brewer-Homebrew-Thermowell/dp/B01KP4D0K2/ref=sr_1_1?ie=UTF8&qid=1484750035&sr=8-1&keywords=thermowell