It does record your choice in the addon options and starts logging, the prompt just stays on the screen.

I haven't tried pressing no, as I've only done a few new dungeons, and they were on difficulties I wanted to log.

I'm also not 100% sure it isn't caused by another addon, as I had a couple errors pop up when zoning in

Hi - thanks for the report -- I have been using in both mythic and heroic dungeons in DF without seeing this issue. If you can reproduce with other addons (especially those triggering errors) enabled, please reopen and also include one or more zones where you see it occur.
Thanks!

I finally decided to go back and do some test cases because I got tired of the r.io addon doing an inconsistent job of enabling logging. I can't reopen this ticket, but if you don't end up seeing this, I'll create a new one.

I used Ruby Life Pools and Algeth'ar to test. When I zone in without LoggerHead, no errors. When I zone into a new difficulty for the first time, I get the issue described above, where the popup doesn't go away when clicking on one of the options, and clicking the Enable button causes the following error. I also get this error when I zone into a dungeon where I've enabled combat logging via LoggerHead:

1x ...eBlizzard_CombatText/Blizzard_CombatText.lua:448: attempt to compare two nil values
[string "@Blizzard_CombatText/Blizzard_CombatText.lua"]:448: in function <...eBlizzard_CombatText/Blizzard_CombatText.lua:429>
[string "=[C]"]: in function `CombatText_AddMessage'
[string "@LoggerHead/Libs/LibSink-2.0-100000/LibSink-2.0.lua"]:399: in function `func'
[string "@LoggerHead/Libs/LibSink-2.0-100000/LibSink-2.0.lua"]:532: in function <...eLoggerHead/Libs/LibSink-2.0/LibSink-2.0.lua:518>
[string "@LoggerHead/Libs/LibSink-2.0-100000/LibSink-2.0.lua"]:538: in function `Pour'
[string "@LoggerHead/Loggerhead.lua"]:325: in function `EnableLogging'
[string "@LoggerHead/Loggerhead.lua"]:306: in function `Update'
[string "@LoggerHead/Loggerhead.lua"]:242: in function `ForceUpdate'
[string "@LoggerHead/Loggerhead.lua"]:276: in function `accept'
[string "@LoggerHead/Loggerhead.lua"]:103: in function <LoggerHead/Loggerhead.lua:100>

Locals:
message = "Combat being logged to Logs/WoWCombatLog"
scrollFunction = <function> defined @Blizzard_CombatText/Blizzard_CombatText.lua:644
r = 1
g = 1
b = 1
displayType = nil
isStaggered = false
string = CombatText1 {
 scrollTime = 0
 scrollFunction = <function> defined @Blizzard_CombatText/Blizzard_CombatText.lua:644
 0 = <userdata>
}
noStringsAvailable = nil
yDir = nil
lowestMessage = nil
useXadjustment = 0
(*temporary) = nil
(*temporary) = nil
(*temporary) = 1
(*temporary) = <userdata>
(*temporary) = nil
(*temporary) = nil
(*temporary) = nil
(*temporary) = nil
(*temporary) = "attempt to compare two nil values"

Since the error seems to do with letting me know that LoggerHead has enabled combat logging, here's a screenshot of my LoggerHead - Output options panel:
https://i.imgur.com/8rnEGAd.png

After making sure that was all correct, I then chose one of the output options and there's no longer an error. I'm not sure how it got into that state (I don't remember uninstalling anything that was a valid output before; I'm pretty sure I've always had it set to None because I rely on an LDB display to know if it's enabled or not), but there's no problem now.