The simplest thing you could try today is to warm the room, and the toilet seat itself, if you can find a way to do that! He may be ultra sensitive to the cold.
Has your dad been examined by the GP to check whether there is a physical problem causing him pain? It could be an impaction as you first thought, but could also include back pain or muscle problems that make the action of sitting down on the toilet painful.
I would recommend that you ask for an urgent OT (Occupational Therapist) assessment via your dad's GP. You could get from them a portable toilet seat frame on loan to try out.
It is also possible to get a wheeled commode that can either be used with bowl in place, or without bowl positioned directly over the toilet. If your dad is still frightened of the toilet when he uses a frame with arms he might be less frightened if the commode bowl is in place and he can't see the toilet bowl.
Many people with dementia develop a fear of water and a distorted sense of distance. Small children can be fearful of the toilet because they think they will fall in and be flushed away. It's hard to say whether your dad is frightened of the toilet because he is unsteady and might fall, or he thinks it will swallow him up, or because he can no longer 'see it' because it is white (a red toilet seat can help with this), or whether he has a fear of letting go of his poo (again this is a fear some children have because they think they are losing part of their body, not a waste product that won't harm them if it comes out).
Lots of suggestions, I hope it has given you more ideas. Very best of luck.