The Remington 870 is probably this forum's favorite shotgun. It's a very good shotgun, and is commonly issued to police. The Wingmaster and Police models are better constructed and finished. The Express models are more roughly finished and have some plastic and more cheaply made parts. It has a crossbolt safety, and loads with the carrier down, which I find obnoxious.
The Mossberg 835 is kind of an unusual option. If you don't have any particular use for a 3 1/2" shotgun, you would probably be better served by an ordinary Mossberg 500, which is also a very good shotgun. The Mossberg 500 is a little lighter than the 870 since it has an aluminum receiver, it has a tang safety, and it loads with the carrier up.
I'm not terribly familiar with the Winchester 1300.
The Browning BPS is really a beautiful shotgun. If you are left handed, this would be the best option. It has a tang safety, ejects shells straight down, loads with the carrier up, and is super smooth cycling. It's probably going to be smoother cycling than the other shotguns you mentioned.
Another really good option would be the Ithaca 37. Out of the pumps I've fired, it's my favorite. It's similar to the BPS, which was based on the Ithaca 37. The Ithaca is lighter than the BPS, is also very smooth cycling, and has a crossbolt safety.
Noyes is correct in that you should see which shotgun fits and feels the best. It really does make a tremendous difference. Also, make sure you try out the controls and see what works and doesn't work for you.
For example, I have small hands, which makes a lot of slide releases difficult to use, and I don't much like tang safeties. There's nothing wrong with tang safeties, it's just a matter of personal preference.