My assumption has always been that anything boosting Luck in BL3 worked exactly the same way as the Vault Hunter’s Relic in BL2. That is to say, it alters the weight pro rata for different rarities when rolling for an item (eg world drop loot pool), but that wouldn’t necessarily increase legendary drops significantly (unless they were already boosted by other things) and in any event was very unlikely to affect designated drops.
The Lootsplosion mayhem modifier is an interesting one: it basically increases the total number of rolls per enemy. If that enemy is higher tier, then you automatically get increased odds of a legendary dropping out (and it may increase Luck as well?)
So if you want lots of orange, play on maps with lots of badass and above enemies. Some of the trials maps are particularly good for that. If you’re looking for a designated drop, though, good luck.