I don't think there is a way of booting into DS mode with the Omega, certainly not an officially sanctioned one and you do risk something unpleasant happening there if you are not careful. The older EZ4 kernels and whatnot had DS loaders on them so you could launch into it with a flashme modified DS or a nopass when the thing was still in DS mode with all the DS hardware. At that point you could then launch your chosen homebrew (such things also had a DLDI patch so you could write to the SD card) which mostly would have been save and ROM dumpers after DS flash carts took off.
The 3 in 1 worked a bit differently and instead it used the GBA functionality to grab DS games 32 megs at a time and stitch it back together later.
If you don't have a homebrew capable DSi or 3ds and don't want to get an old flash cart, save/ROM dumper (there were a few towards the end of the DS lifetime and a few others from the 3ds that also worked here) or the like then your best bet is probably going to be wifi based.
Various tools exist that allow you to either FTP or simply beam (think one used netcat if memory serves).
Today most would suggest just getting a modified 3DS and using that.
You will however need a compatible router and to set it to no security or WEP mode (that is a hardware limitation) as the dumper program requires. It might also technically not dump the DSi enhanced and the later signed games properly*, though outside of having to disable a few checks on ROM hacks or being able to use the DSi stuff in DSi mode then not in any way that matters.
*early on it was thought an area of the header was blank, and it was. When the DSi hit and got some more security (
https://hackmii.com/2010/02/lawsuit-coming-in-3-2-1/ ) all the DS games from then on included a signature in the previously unused area. No DS emulator, flash cart maker, DS device clone, ROM hacker, emulator author, flash cart user... will likely ever care about it but it was the reason we saw a lot of games redumped over the years following the last DS games from groups like PUSSYCAT.
https://web.archive.org/web/20180920175520/http://www.abgx.net/nds_releases_date.txt http://www.advanscene.com/http://www.advanscene.com/ if you are bored.
like a 250 dollar kit back when DDR4 was Intel only
