I found that targeting the swiper scrollbar CSS caused knock-on effects with the translate-3d that Swiper had already calculated, so the best solution I found was just setting it via the wrapper:

<!-- Custom Scrollbar -->
<div class="w-md max-w-4/5 mx-auto relative top-6">
  <div class="{{ $uniqueId }}-scrollbar swiper-carousel-scrollbar"></div>
</div>

This is tailwind CSS but you get the idea - set the desired width on the wrapper, then max width 80%, center it (margin: 0 auto) and set to position relative since the scrollbar is absolute, and move down slightly so it's not hidden under the slider.

This has worked a treat.