missing some specs here, or at least photos. what are the letters made of?
my first thought is to simply add new studs, as stated above. the pattern is already drilled-into the wall so re-mounting should be a breeze. if holes are filled with old broken studs, then rip them out.... no, you do not have to drill all the way through the wall unless your intent is to bolt it from the backside, a process reserved for much much heavier construction applications. 1-2" into the brick is plenty.
if adding new studs to letters is outside of your scope, then how do you intend to make a new pattern? however you do, then mark a bunch of spots at ~10" spacing around the perimeter, get some 1" angle iron and cut some 1" tabs & tapcon drill them into the brick, cap it with the letter & secure with screws through the return of the letter.
if you're hoping to simply tape/glue these to the building..... stop right there. never ever think that way again.