The only known way to hack a 11.3 3DS to run CFW is by using sighax, though it's possible that other undiscovered methods exist. But for that to be a possibility, someone would have to develop a tool to dump the bootrom, then use the findings from the bootrom dump to develop an implementation of sighax, which could then be used to install CFW on any version console. But realistically you are going to need a hardmod to install sighax, and they cost around $/€70 unless you are or you know someone who is proficient enough in soldering to do it.
But we have actually known about the bootrom dumping method and sighax for a while and there seems to have been no progress on actually developing something that makes use of them. It's either extremely complicated, or there's just not that much interest in 3DS development anymore. Either way, it seems like it could take a long time and might never happen.
This is why everyone says to sell the 3DS and buy another on a lower firmware in threads such as this, it really is the cheapest and quickest solution, and the only guaranteed "fix".
And that's my suggestion to you as well. You can back up your saves with JKSM before selling your console, and put the money towards a new console, or if you don't want to spend any extra money you can buy one used with <=11.2. Once you have received the new console and hacked it, you can contact Nintendo to get your NNID transferred either over the phone or by using this form:
http://en-americas-support.nintendo.com/app/ask/ht/520/cp/593/mq/515 (Note, not sure they will help you through that form seeing as it's on the American site), you'll just need both your old and new serial number.
That way you don't lose any data in the process.
@
Psionic Roshambo:
like a 250 dollar kit back when DDR4 was Intel only