I really like the sound of the BFF on but I heard the anarchy does lots of damage. I just want my robot out for a long period of time
Well to keep out DT for as long as you need him, you need to invest into BFF significantly. Normal can get away with you just getting down to Upshot Robot so that kills extend his duration time. TVHM on the other hand is going to force you to invest into other trees, such as LBT for the shock damage and other skills. BFF also needs a more significant investment due to DT needing more survivability, and therefore forcing the need to have Sharing is Caring so that your shield becomes DT's shield as well. This also begins to affect your gear choices since it benefits you most to share Roid Shields (ie Love Thumper), and Nova Shields (ie Flame of the Firehawk.) Antagonist is another shield that is good for sharing due to its effect to slag on any hit makes battles easier in general, and the massive damage from reflected bullets isnt a bad perk on the side either.
Now if you are going to use Anarchy, there really is no need to invest past Discord, due to that being more of a safety net to protect your stacks from errant reloads, and make death the only way you are going to lose them all if you intend to keep your damage bonus intact.