Do you free feed him or is it scheduled feedings? If you free feed I would consider taking away food at about 5pm. I did that to Kaos when he was a pup and he stopped having accidents in the crate at night. Also, how big is the crate? It should only be big enough for them to stand up and turn around in, other wise they are more likely to potty inside because they don't have to be close to it. Also have you considered putting a potty pad down for him to use? The fact that he is pottying on the door mat, makes me think he was potty pad trained previously, and might be the easiest thing for him to transition to. Oh and Daniel isen't the only yorkie that doesn't like to do his business outside when it's cold or wet. I think most do not like to. Kaos HATES cold/wet weather, so for bad weather I will put a potty pad down for him. (even though i prefer him to go outside, its better than accidents on the carpet)
Good luck! I hope this helps.