I assume that's a clear coat of sorts(oil-latex),perhaps a satin finish.
You can dull the finish with some fine sand paper and re prime it if it's an oil clear coat.
If it's latex/water base clear coat,you can go over it with latex paint
latex won't stick to oil but oil will stick to latex.
The inside of the slats might be tricky and you would want to cover the fabric that's in between the slats,
perhaps painters tape slid underneath each slat or even drywall tape as it's very tough so no dust or paint gets on the fabric.
We use it under base boards(new housing)when the owners have semi shag type carpet as the fibers are quite long and stand above the base somewhat,we tuck it underneath the base and it pushes the carpet down so we can paint the whole trim surface.
No fun trying to cut in around thousands of shag fibers

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