How do I find the cutting point on neon flex with no marked lines?

The dark LED repeating on a fixed count is actually a decent clue, but I wouldn't bet the whole strip on it yet. On some neon flex the manufacturer leaves one position dark at each segment joint, so a dead-looking chip every 17 LEDs lines up with the idea that those are the section boundaries. But it coud also just be one failed chip that happens to sit there, so confirm before you cut. Here's how I'd check it. Power the strip off and shine a bright phone flashlight through the silicone from behind, right at one of those dark spots. If it's a joint, you'll usally see the copper inside pinch down to narrow pads, sometimes with a tiny gap between sections. A single dead LED looks diffrent, the copper runs straight through uninterrupted and the chip footprint looks the same as its neighbors. You can also run a fingernail along gently and feel for a slight ridge or step where the internal sections are joined. Then do the low-risk version of the cut. Leave your 3 meter section untouched and sacrifice a piece from the far end of the 10 instead. Cut right at one of those dark spots, power everything back up, and check that the whole remaining run still lights. If it does, you've confirmed the cut point and can measure out your shelf length with confidence. If the section past the cut stays dark, back off one LED position and try the next dark spot.

One more thing once you make your real cut: seal the end. That silicone sleeve is what keeps moisture off the copper inside, so push on an end cap with a bit of adhesive, or even a dab of hot glue inside will do for an indoor shelf. An open cut end will corrode on you eventually if you leave it bare.

The idea of testing a small piece first is smart because I don't want to mess up the whole strip. I am worried about using glue or caps on the end since those parts might get hot when the LEDs are running. If the power gets too hot right at the cut, I wonder if the adhesive itself could be a fire risk or if I am messing up the insulation. It seems like adding any material could change how the current flows through the section. How much physical coverage do I need to make sure I don't cause a short circuit?

I totally agree with the idea of using the flashlight to check the internal copper. That method is definitely the least risky way to confirm if you are looking at a joint or just a bad LED. When I have done this on other strips, I've found that if it's just a chip failure, the two copper lines leading into the dark spot are perfectly straight. However, if it is actually a cut between sections, the internal wiring usually shows a sharp kink or bend right at the point where the two sections are joined togehter.

I ran into this exact issue a while back on a retail sign project and honestly, the dark LED thing is a gamble. What worked for me was just looking closely at the solder points under the sleeve. Even if you can't see a gap, the copper connection usually looks thicker or different right at the seam compared to the uniform flow you see between chips. You are probably better off checking the solder integrity than counting dark bulbs. If the wire looks like it joined a different piece of copper, that's your line. I found that a few cuts that were one chip off resulted in noticeably uneven heat distribution under my heat gun setup, which is another giveaway.

Update: got it sorted. Ended up using the flashlight trick from the replies, holding a bright light behind the strip to see the copper inside. Sure enugh, the repeating dark LED lined up with a wider solder point and a small gap in the traces every 17 chips. Cut at one of those spots on a scrap end first, lit up fine, so I trimmed the shelf length with confidence. Worked a treat.