Firstly, if you want a dropped kerb, decide where exactly you want it and what size, and apply. Don't let an neighbour dictate what you should do.
Secondly, how much of your garden are you wanting to make into your drive? Half or all of it? Looking at your picture, I reckon you could get away without having to move the lamp post if you put a standard width dropped kerb to the right of it as you look at the picture (where the truck is). I know it might be a bit fiddly moving cars around, but you might get used to it. If it did turn out to be a problem for you, you could always get the lamp post removed later and have your dropped kerb extended.
Your neighbourooney can't complain about the planning app really, as your not affecting anything. I went through the same application as I'm on a main road, and I had to do a design and access statement and just went on about how it will be good for road safety etc blah blah
You have every right to apply for and have a dropped kerb in front of your own house, it's the other guy's problem if he's got no where to park his truck.
As someone said before, you may not need planning permission if it's not a a, b, or c class road. You may need planning permission to construct a driveway however. Most local authoritys (particularly in high risk flood areas) are requiring planning consent as more and more people want to remove grass and gardens to construct hard standing driveways. You may need to check this, if you haven't already.
As for people saying you can't park trucks like that on residential roads, why not contact the council (anonymously if you want) and complain. Say it's not allowed and refer to whichever highways regulation it is. Say it's very dangerous for this size vehicle to be regularly parking there because of children playing etc. I'm sure they'd investigate it
Good luck!