Oh, that’s a really interesting question!
As a general note, I think if your recall is good enough, then this shouldn’t be a problem because you will be able to recall enough of the early-night-LDs even if you don’t take notes or anything. I almost never take notes when I wake up and plan to go back to sleep, I only write down at the actual start of the day.
But this doesn’t answer your question of course. I think the discussion around the answer to the question can get quite philosophical. If you don’t remember what happened inside the dream, did it really happen? Or rather: What’s the difference between a dream you don’t remember and a dream you never had? If, according to your personal opinion, there is no difference, then you should probably take notes and accept the lower chance for follow-up LDs.
On a more practical point, you know at least that you had the LD, and it probably also had some effect on you. It will have at least shaped your mind a little and influenced your psyche to some degree. (According to some/many scientists, dreams are a threat simulation device to better prepare you for the real world, and LDs would serve this purpose even better plus additional situations that also prepare you for the real world.). So if you believe that this is true, then you should go back to sleep instead of taking notes.
There is also a different result-based view that you can assume. Let us assume two scenarios.
First one: You have an LD, remember it well, take notes but because of that, you fail to induce a second one in that night. Result: 1 well remembered and documented LD
Second one: You have an LD, remember it after waking up but don’t take notes. You immediately go back to dive back into dreaming. Your course of action makes you forget the first dream but you are having another LD. After waking up, you remember the second LD well and take notes of it. Result: 1 fuzzy recollection about an LD plus 1 well remembered and documented LD
If you compare these two outcomes, it seems that scenario two is slightly better. The only risk is that you may fail to get the second LD, so that’s what you should contemplate when making your choice.
There is another point to consider that would almost always decide the course for me: Sleep quality. If you wake yourself up more than necessary so you can take notes of a dream, maybe your subsequent sleep will have poorer quality or be too short, if you lack the time for the sleep break. Depends on your RL situation and not your dreams, but I think you should pay attention to this, too.