I'm seeing a lot of connected things here.
To put it bluntly, if your HDR grades are better than your SDR grades, that's just a limitation in your skill level of grading. I say this as someone who took an embarrassing amount of time to learn to colour grade myself, and even now I still feel like I'm not getting the results I'd like.
But this just goes to reinforce my original point - that one of the hardest challenges of colour grading is squeezing the cameras DR into the display space DR. The less squeezing required the less flexibility you have in grading but the easier it is to get something that looks good. The average quality of colour grading dropped significantly when people went from shooting 709 and publishing 709 to shooting LOG and publishing 709.
Shooting with headlamps in situations where there is essentially no ambient light is definitely tough though, and you're definitely pushing the limits of what the current cameras can do, and it's definitely more than they were designed for!
Perhaps a practical step might be to mount a small light to the hot-shoe of the camera, just to fill-in the shadows a bit. Obviously it wouldn't be perfect, and would have the same proximity issues where things that are too close to the light are too bright and things too far away are too dark, but as the light is aligned with the direction the camera is pointing it will probably be a net benefit (and also not disturb whatever you're doing too much).
In terms of noticing the difference between SDR and HDR, sure, it'll definitely be noticeable, I'd just question if it's desirable. I've heard a number of professionals speak about it and it's a surprisingly complicated topic. Like a lot of things, the depth of knowledge and discussion online is embarrassingly shallow, and more reminiscent of toddlers eating crayons than educated people discussing the pros and cons of the subject.
If you're curious, the best free resource I'd recommend is "The Colour Book" from FilmLight. It's a free PDF download (no registration required) from here: https://www.filmlight.ltd.uk/support/documents/colourbook/colourbook.php
In case you're unaware, FilmLight are the makers of BaseLight, which is the alternative to Resolve except it costs as much as a house.
The problem with the book is that when you download it, the first thing you'll notice is that it's 12 chapters and 300 pages. Here's the uncomfortable truth though, to actually understand what is going on you need to have a solid understanding of the human visual system (or eyes, our brains, what we can see, what we can't see, how our vision system responds to various situations we encounter, etc). This explanation legitimately requires hundreds of pages because it's an enormously complex system, much more than any reasonable person would ever guess.
This is the reason that most discussions of HDR vs SDR are so comically rudimentary in comparison. If camera forums had the same level of knowledge about cameras that they do about the human visual system, half the forum would be discussing how to navigate a menu, and the most fervent arguments would be about topics like if cameras need lenses or not, etc.