If I print on double sided glossy photo paper and add a glossy laminate, is the finished card recyclable as it is, without separating the layers?

No. Photo paper is a plastic coated paper, and neither photo paper nor lamination film is typically recyclable. Both can contaminate paper recycling streams, so we would not recommend that combination for business cards or backing cards if sustainability is the goal. For business cards we would suggest using a printer that specialises in board or uncoated cardstock, with eco-friendly inks if you want to go further.