A bias cut is usually done with fabrics with good drape, linen is not a traditional fabric for that. A bias cut is a very inefficient use of fabric (it's cut at a 45 degree angle to the fabric edge) so it's much more costly and requires more skill to sew so the prices are usually high and you rarely see it on the high street.
Liliysilk will have some silk dresses at around your budget but not sure if they're bias cut (I suspect not). But not sure with Brexit if you're paying import duties and VAT now.
Raey (at Matches Fashion) have some nice silk dresses too starting at around £140 IIRC, don't think they're bias, although the more expensive ones may be.
Samsoe & Samsoe do utilise bias cut in some of their skirts and dresses, sometimes it's just the skirt part but they use a lot of cellulose based fabrics which also have good drape, should be around your budget too.