All store-bought lard will have additives such as BHT to preserve flavor and deter rancidity. Home rendered will not have additives and the additives do not make a difference in making soap. False trace really does not happen when using OO, it happens when using hard oils such as lard, tallow, palm, soy wax, and butters when soaping too cool. What happens to the hard oils, if the lye solution is room temp or cooler will cool down the oils too fast causing them to start to solidify or thicken sooner than normal. When this happens you just have to get a spoon and stir until your lye solution starts reacting and the oils heat up, you will notice a color change in the batter and it will start thinning out, then just proceed as normal.
Oh sorry, it looks like I was unclear. I did not mean the OO caused the issue with false trace. I meant that it was a combination of 2 issues that caused me to have some issues with that batch. I used palm kernel oil, no-stir palm oil, and coconut oil as well making up about 58% of my oils and soaped in winter on a much colder day than my previous attempts.
Since I was unaware of false trace at that time, I did not continue mixing once it started to thicken up, and simply started mixing in micas and olive oil and pouring my soap batter into my mold. I figured out the false trace issue the day after I made that batch because I'd wondered about the extra oil and how quickly the batter had thickened up that day. If I recall correctly, the surface temperature of my batter had already dropped below 80F (26.67C) at that point in time.
I had not heard of false trace at that point but thought something was weird with how fast things were cooling and tracing and the slightly grainy texture of my soap. The room temperature was in the 60s (Fahrenheit this is roughly around 15-20C) and my usual attempts were made in a room that was about 10F or ~4C higher.
The olive oil issue, is because of things tracing faster than planned, I went for coloring as quick as possible (yet another mistake) and accidentally poured too much olive oil into my micas. At that point I was just trying to be as fast as possible, and hoping luck could save this batch. Then I oven processed it.
It came out okay, eventually. But I nearly wrote it off entirely and threw it out.
As for why I thought the lard's source might be part of it:
I've seen store-bought lards that have citric acid in them. Based upon what I've read, I figured that citric acid would typically react with NaOH and form sodium citrate which can use up some of the lye as well as potentially skew some other additives. Unfortunately, I also couldn't say how much citric acid is added to the lard and if there would be a significant enough amount to affect anything.