when i use custom bars
Overflowing Bars [v21.4.0],
Detail Armor Bar [v2.6.3],
Tough As Nails [v10.4.0.3]
they dont get moved with the hotbar

on Raised [v.4.0.1]

Your issue seems to only occur on Fabric when Overflowing Bars is installed. It seems this mod did support an older version of Raised, pre-4.x (which last got a release for 1.20.4), which only attempted two specific single-component 1-dimensional translations. Raised no longer supports the old Object Share objects because of the increased complexity of moving to multi-component 2-dimensional translations. In short, this is because Raised added support for translating on the x-axis, and expanded to 8 different targets.

However, 5.x is looming around the corner and it will not support the 4.x API, so I would hold off on mentioning this to OB until then. You'll be able to add in support for your own specific layer instead of the 8 vanilla targeted layers.

Can confirm this issue is still present on Forge 1.20.1 using Raised 5.0.1 with Overflowing Bars. Is there a workaround to this or should I downgrade to Raised 3.1.3?

Until OB replaces the Object Share way of using the now non-existent Raised object with a call to Raised's API to get the X and Y offset and register a layer, yes you would have to.

You can always make a small patch mod to mix into OB and do that yourself in the meantime. The wiki offers an example of how to add support for the modern 5.x way. You can either register a new layer for it (e.g. "overflowing_bars:bars") and get the values for that or just use an existing one that Raised has created for the vanilla stuff (i.e. "minecraft:hotbar").