Also try Hainsworth in the UK.

Their doeskin is a perfect match for the weight and texture of antique case lining. https://www.hainsworth.co.uk/collections/doeskin/

It is strong, but thin enough that it doesn't add ugly bulk that you will often see with relining projects where an unnecessarily heavy baize was used.

Get some samples from them as well to compare.

“I left long before daylight, alone but not lonely.”~Gordon Macquarrie