My dizzy-rt branch https://github.com/htot/meta-intel-edison/tree/dizzy-rt builds not only a kernel, but also U-Boot and rootfs.
(dizzy is the Yocto version, with kernel 3.10.17-rt, old wiki instructions here: https://github.com/edison-fw/meta-intel-edison/wiki).
You should be able to just flash that (using flashall). Presuming that it downloads and builds - and I haven't built that in a long while. If you build this you really need Ubuntu 14.04 (not later), else it won't build.
As you mention "the dizzy-rt branch (meta-intel-edison) fails in some recipes as the sites are no longer available" we could try to remove those recipes, if you are not using them. But I am afraid java is going to be a problem.
I understand you would prefer to just build the rt-kernel (from here GitHub - htot/edison-linux at edison-3.10.17-rt ) and replace the existing. Normally that won't work, because you have kernel modules on the rootfs built for the other kernel (under /lib/modules/..).
But of course, you can build not only the kernel, but also the modules and install the new modules manually on the rootfs side/-by-side with the old ones. That should work. You need a relatively old gcc to build such old kernel else you'll get many warning and errors and the build won't complete.
Complicating factor: if the kernel doesn't work you can't boot Edison and you need to recover old factory image before you can continue. Probably better idea is to add -rt kernel as extra kernel, and keep the original, until you are sure it boots. And to boot the -rt kernel, interrupt U-Boot, then run the macro start -rt.
My preferred solution, we have newer Sumo version, with recent kernel 4.18.0, new U-Boot, acpi tables for configuring hardware. This actually works very nice, except maybe we dropped a few things (some cloud services don't exist anymore).
The old 3.10.17-rt kernel was not very real time, the last I tried (4.16) worked much better.
In your case, you want to use mraa and upm. Mraa is not yet working with the acpi enabled kernel., so I would suggest non-acpi version of sumo32 (32 bits) and modify to use 4.18.12-rt7 kernel. I can help you with that, it's not that much work.
Basically we need to apply the -rt patch on 4.18.12 and rebase edison kernel patches on top of that and then fix the recipe to use that.
Nice side effect, we have a much newer mraa and upm in sumo, see What's in the rootfs and the initramfs images | Edison
My most up-to-date documentation of sumo32 is here: https://htot.github.io/meta-intel-edison
The bad news: you will be needing java bindings for mraa, and we don't have that currently. I think they are not even supported by mraa at the moment. We have bindings for python, nodejs, C and C++ that I just recently tested with a I2C device and work. So if one of these is an option for you I expect sumo is the way to go.