I'm knocking on wood while I type here as I don't want to jinx it again, but it appears that Elijah's sleep may be improving!
The past few nights had been marginally better and I'd managed to get him to nap during the day (albeit, still mostly short, fretful naps, but naps nonetheless). So he was more rested during the day, which enabled him to relax a little more at bedtime. We've also tweaked his bedtime routine so it starts earlier and ended with feeding him to sleep (but removing the boob from his mouth before he went right off to Snoozeland). Yesterday, my pal Angie, an osteopath, took pity on us and came around to treat Elijah. She will do a follow up on Friday. This seems to all have helped.
Last night, because I'd managed to resettle him for a longer nap in the afternoon, we decided it was time to try the cot again. Our fallback position was that the first time he woke during the night and didn't settle without being picked up, he would come back into bed with us. Well, the good news is that after fighting sleep (because the feeding to sleep didn't work with the transfer to the cot) and finally getting to sleep around 8pm, he slept until 2:25am! Hooray! The bad news was that he wouldn't resettle, wouldn't feed to sleep (he fed, but was unsettled throughout), wouldn't be cuddled, wouldn't lie on us. I tried feeding him again and he must have been so exhausted this time that he eventually fed to sleep. This whole process took at least 90 minutes. However, he didn't wake again until right on the dot of 6am, so we are definitely making progress.
At the moment, he's having his morning nap. He has been down for about an hour and a half and barely needed a resettle at the 20 minute mark, which is usually the point where he wakes up and won't sleep again.
So I think we have a breakthrough, but I'm still knocking on wood.