I've been buying my first headless 34 years ago and since then have been owning and playing hundreds, really, of headless basses.
I never had a problem with the clamping system as long as you cut the strings after clamping them with the needed pressure of the hex screw.
I've put some White Nylon's on my Leduc U-Basse 4 fretless and simply noticed that the G string needed more screw pressure when clamped.
After that it was staying in tune perfectly and I could detune it and retune it without any problem.
Before that, some headless fretless basses have had Black Nylon's without any issue at all, even my sixers.
Check your clamping screw, the clamping side must be slightly flat rounded (slightly flat ball shaped to match the pressing point) without any asperity that could cut the string.
If you have this kind of hex screws, you are in big problem, so round them much flatter and it will be fine :