when i put in the doubled up 2"x12"x20' lvls in the new dining room addition, my engineer had me use 1/2" bolts thru every 2' OC staggered up and down the run at 2 1/2 inches from the top then bottom. i should think nails would allow too much shifting.
i agree with bob to have an engineer/architect do the math, but if you cannot do that, then maybe just one more lvl bolted tight will stop the movement? seems to me steel would be quite expensive, (as if lvl isn't! lol)
and heavier and harder to work with. i think the bolts will help stop the movement. perhaps you could try them alone first (get them 2" longer than needed) bolt what you have tight and see if it helps, if not enough, add the extra lvl and rebolt and see? i am not a contractor/builder or anything, just trying to help from my own experience. hope this helps!