Two versions: 

1. Button Lowered Weapons.esp - Weapons stay lowered when you leave cover. No issues.
2. Button Lowered Weapons - Auto Raise from Cover.esp - Weapons raise back up when you leave cover. I had to use movement conditions to get this to work properly (or once you lowered your weapon you'd go into cover and it would raise back up). So if you run into a wall and keep running for a second your weapon may raise up. Aim out of cover once and it'll go back properly. If you are in cover and then spin around while otherwise standing still (without moving) your weapon won't raise - I had this fixed but then spinning into cover would raise the weapon, so I picked the lesser of two evils.

Installation - Install the .esp you want along with the .esm file that comes with it into your data folder. Make sure they are both activated. The master file is called Arbitration - Resources.esm  - I plan to use it for aspects of my gameplay mod. It works here to add the walk speed fix as well as keep you from having trouble if you switch from one .esp file to the other, or if you decide to remove the mod. It is a required file, and your game will not start without it enabled as a master.

Q: You make mention of walk speed bugs and such that weren't ever in this mod. Wassup?
A: There was once a mod that I just killed called Controllable Lowered Weapons. It was my mod. I shot it in the face with this mod. I had planned to keep the two separate but even though I used different functionality (just a game setting in the prior mod) the main file in this mod replaced that mod by being more better. I realized this when I went to upload the 2x cover distance - there wasn't a slight yet perceptible difference, they were one and the same.

Q: I was mostly wondering about the walk speed bug, not so much about your life story. WTF was with the walk speed bug?
A: I was all happy and superexcited when I figured out a way to keep a weapon lowered based on where a player was looking (prior mod). It wasn't long until some people came along asking about why the player character looked like he/she was on amphetamines while walking with the weapon lowered.

Q: But walking with the weapon lowered works for the superfamous lowered weapons mod! Why were you so incompetent?
A: I went to look and sure enough Bethesda had no modifier for the lowered animation at walk speed. Having tried the superfamous lowered weapons mod I figured it would work fine. But the superfamous mod had simply renamed the lowered animations to the raised animation names so that the game thought it was playing these animations - and you never had any raised animations, but the lowered animations got the reduced animation speed that was being applied to the raised animations. Actually having the animations lowered resulted in an animation that played too fast to be the animation of a person holding a lowered weapon while walking.

Q: So why didn't you just learn to be an animator-person?
A: I delved into trying to modify 64-bit Havok animation files. Even with the right tools, handed out freely by Havok before they decided not to hand them out any more, the behavior files could not be edited since Bethesda had altered them for their own needs and all tools from Skyrim that had dealt with this (just one tool, really) were 32 bit. 

Q: So you fixed this how?
A: After a stupid amount of time that one wouldn't think necessary, I stumbled across the animation speed modifier in Fallout4.exe (was looking in the .esm the whole time). This doesn't show up in FO4Edit when you're looking for things to mod since it's not in the .esm file. I had, in the meantime, figured out how to adjust modifiers like this based on certain game settings, so I figured out how to slow the animation down when you're walking with your weapon lowered. So now, in spite of the fact that the game sometimes takes half a second to figure out what you're doing, the walk speed bug is fixed. So now I strut to the mailbox in the afternoon when the mail comes, and my mailperson has no idea why.

Q: Do you really look like John Travolta?
A: No. My ass looks way better than his.