I wouldn't advise using weed supp. membrane under turf as it will stop the grass roots from getting down into the soil and it will curl up and die in the first dry spell of weather. Buy some weedkiller that contains Glyphosate and apply this as instructed on the packet. This type of weedkiller becomes inert on contact with the soil and won't be around to harm the turf.

But don't apply the glyphosate weedkiller to the turf as it kills grass as well.
It would be better to treat the buttercups first. The weedkiller is absorbed through the leaves, so spray it onto the weed leaves and let it be absorbed. Wait at least until it's dry before laying the turf.

ideally you need to wait til the buttercups and other weeds are growing well, then treat them with glyphosate weedkiller and leave for a few weeks. If you already have the turf, I guess you'll have to lay it sooner than that.

I'd systematically and rigorously dig out the buttercups first - tedious work, removing all the roots, but will give a quicker, better result.