After investigation, detecting the toolbox while in remote view seems to not be easily possible due to an engine limitation. I also don't mind it for thematic reasons (gotta concentrate on what you're crafting) but I did make an attempt to implement it as a setting, without success.
While crafting an item, the speed bonus is applied normally to the weapon slot, but if you cycle through the weapon slots during the process, the bonus is lost.
That sounds like probably intended behavior. You need to have the Toolbox in your hands to use the tools.