I setup a cheap 12v car style battery (from a mobility scooter) and a USB night vision camera in a cardboard box to help a farmer friend who had a fly tipper. Worked well enough to give the evidence to trading standards / Council and the police. (Police couldn't do anything unless there was a witness, and even then it's a civil offence). Might be useful for you to consider. Maybe even put the camera under the car and use the car to power it?(Don't know if that would work).
Oddly leading dog poo on private land is very similar, it's classed as "rubbish on private land", which is a civil matter.
If your council is lazy, they might start to look at "mediating", or if they are half decent, they will give a caution out to the dog owner, and if they are decent, they will give a Fixed Penalty notice to them. They can only do this if there is a clear picture from the cameras, I.e. don't cheap out on cameras.
You can take the dog owner to civil court, but I'd suspect that it won't change anything.
Just to set your expectations, the police will likely only offer to have a word with that person at best, because technically it's a civil matter. Try to make sure the camera only picks up things on "your" property, or IF it picks up anything on public land you must display a prominant sticker saying that you have a working CCTV system. Otherwise the police may have to tell you off for invading someone's privacy.
If you were living in fear from an abusive neighbour who was repeatedly harassing you, then the police would be able to take it a bit further perhaps. I wonder if defacement of property (I.e. the door) might be able to be said to be abusive.. I don't know.
You will need to keep the cameras there whilst you live there or that person doing it moves from the area. Therefore you need to hide them well or put them completely on display.
I know I sound very negative but you've just got to go around things the right way and believe that it will take time,, but you will get justice. The farmer's case took 9 months for the council to track the fly tipper and get their own evidence, and then 6 more months for the case to go to court. They did win, the fly tipper was given a £5,000 fine and 3 months prison term, suspended for I think 2 years.
GDPR:
Oh and that person has the right to ask for a copy of the recording showing them allowing the dog to crap on your door. You cannot show the images to any of the public, so no posting of it on twitter to embarass that person. You can share with council and police though. They can also ask for the erasur of any pesonal information that you have about them.
If they are the type of person who does this on purpose, they are probably the type who will try to sue you so please follow the GDPR advice.
It's all abit stupid but I think I've covered the technicalities. Apologees if I have missed something.