You may try polyvinylcaprolactam with high molecular weight, having LCST = 32C. Strictly it is not a biopolymer, but its structure is similar to PVP, widely used in medicine.
You might be interested on poloxamers which can be mixed to tune the phase transition temperature, see e.g. DOI: 10.1002/adhm.201600055. Also these are not biobased but regularly used for bioapplications.