Step 1 would be to have a hacked Wii. Pretty much every hacking guide will cover installed hacked IOS modules.
https://sites.google.com/site/completesg/ if you needed a guide for this sort of thing.
I do not know what your general level of code reverse engineering/ROM hacking/coding in general/electronics experience is. It need not be for the Wii but something would help here.
Afraid I have not really kept up with the Wii-IOS hacking behind the scenes/under the bonnet stuff since the very start of it all so I am having a bit of crash course on the matter to type this. You are playing in an area that few people play in, though at least for this initial foray it is not going to be the harder aspects of it. It is possible that you could damage your wii making the initial hacks but if you understand what you are doing and do not try replacing an IOS module used by the system/homebrew channel then the worst that will happen is you have to installed a fixed IOS module.
The IOS runs on an internal ARM9 processor known to most as Starlet,
http://wiibrew.org/wiki/Starlet
That page has docs but I should also note ARM9 is the main processor of the DS and very similar to the GBA's one (and 3ds has some ARM action as well, as does basically every modern mobile phone) so you will find quite a few people familiar with it all. ARM assembly is not that bad and you are not going to need to know all of it by any means.
http://wiibrew.org/wiki/ARM_Binaries has more on the binaries themselves.
I would suggest starting with a hacked IOS and modding that. They do not tend to get distributed as hacked modules any more (rather they are distributed as patching programs) but it should not be hard to dump one and you might still find an already hacked one that works for you. For game running purposes most will tend to have a "base" IOS and install that to a location somewhere (it might be a base of IOS36 but installed as IOS 222 or something). I am not sure what dolphin (the emulator) will be able to do for your for the reverse engineering/testing stages.
It says the IOS modules tend to come and be installed as WAD files (common enough on the wii) and packed within the wads are ELF files (a common executable format).
For WAD files
https://gbatemp.net/threads/wii-and-gc-standard-hacker-grade-iso-and-wad-manipulation-tools.328729/ is what you want.
ELF... it seems most people playing on the Wii IOS side of things tend to opt for IDA (an amazing but very expensive reverse engineering toolkit) where most of us playing on the GBA and DS can get by with the free tools other hackers/devs made. Wii games will not use the ARM processor but the main powerpc one so there is not likely to be much there. I am not sure if the binaries use any compression/obfuscation or if you could feed them into any old disassembler.
Anyway I will have to handwave and assume you have the disassembled code sitting in front of you.
I also looked up the dimming, it is a hardware triggered thing but you got lucky and it is able to be controlled
http://wiibrew.org/wiki/Hardware/Hollywood_Registers (do a page search for dimming)
0x0d80001c is the address it changes and it is only the first 8 bits that are changed.
You want to find anything that fiddles with that in the IOS module. It is enabled by changing bit 7 to high and the luma and the chroma are changed using bits 6-3 and 2-0 respectively, so 8 options for each and I am not sure what the values count as, (personally I would leave the chroma alone so you get the best colours and only dip the luma enough that your overlay device is not fooled any more but you can try that after proof of concept if you want).
Once you have found the thing (possibly things) that writes the value to 0x0d80001c then you will get to work backwards. I do not know how it will be coded here but from the sounds of things there will be a timer that counts up to 5 minutes (the processor has no concept of seconds really so do not go looking for that exactly) and resets with every button press (not sure about stick waggle), and also resets from that mode with a button press. If the code is really good then it will check for the screensaver value to have been checked in the menu (or more accurately passed by the menu into memory when the IOS is called/launched) and only count when the use a screensaver value is there, I do not place much stock in Nintendo's coders but the compiler is pretty good so who knows.
You have many options as to what you can do here. Possibly the easiest would be if the game writes one value to the dimming when it is normal and another when it is on screensaver. Here you would then change the values it writes so dimming is turned on all the time to the value you want. The way I was on about in the previous post would be to get it to ignore the timer and dim all the time -- the timer probably has some kind of compare or check and then a write to the location, you change the check so that it is ignored or always returns the check you want, there are many more options for the assembly hacker but most of those will be obvious once you have figured out what is going on.