To Peter: I can't claim to be an expert on Bugattis, but Mark Dees, perhaps (at least partly) influenced by Conway, stated in "The Miller Dynasty" that "the spark plug cups and valve gear of the Type 51 and 55 Bugattis will interchange with that of a late 91 (Miller)", so yes, "more or less copied the engines" sounds about right. Dees goes on to lament the fact that "Bugatti didn't follow Miller's lead on front end design" in his 4wd car, and else EB appears to have stuck to his own, rather conservative chassis principles. To be frank, the front-drive design didn't promise to be of much use in road racing, and the Miller rear-drives were pretty conservative, too, so there wasn't much to be learned anyway.
To Tim: in 1922, a 4-valve DO engine was no longer "Peugeot/Ballot" influence, it was pretty much state of the art, if not ubiquitous. Miller did have access to several Peugeots (EX3, EX4 and EX5 that we know of for sure) and an S-type Delage during 1915/'16, the time he branched out into the engine manufacturing business, and I'm sure he instantly realized the advantages of that basic configuration. If you've followed the other thread where John Glenn Printz and I discussed the early Miller business (and I'm sure you did), you will know that there are still several mysteries that are nut fully resolved, and maybe never will, but it is my contention that Miller first built a 4V-DO engine that was a very close Peugeot copy for Bob Burman, which didn't perform very well, after which he resorted to a simpler 4V-SO with a lot of Peugeot "DNA", still. After the war, during which Miller had designed and built a 2V-SO aircraft engine, he returned to the DOHC layout and never looked back, except for an unsuccessful "budget" racing head for the Model A Ford in the very early thirties. None of those post-WW1 engines bore any remarkable similarities with the Peugeot, except for that very basic layout, so I don't think one can talk about Peugeot/Ballot influence here.
Tommy Milton's influence in this first straight 8 Miller has, I believe, been overstated over time, not least due to the fact that he was an eager interviewee for Griff Borgeson in the gestation of the latter's "Golden Age of the American Racing Car". I'm sure he did have his say on several aspects of the engine, most notably the valve action, as Miller was always fond of listening to experienced racing mechanics, and taking on even their weirdest ideas - well, there's your Peugeot influence, after all
. Anyway, in his later days Milton always seemed to have an axe to grind, and his famous sentence "I made Harry Miller" is simply rubbish.
To get back to the point, we can see that Harry Miller had always prefered the 4-valve configuration in his performance engines up to that point, so there's no reason to assume that he was "railroaded" or persuaded into doing that. Why, then, the 2-valve '122' and '91' engines? Well, it seems it was just a matter of size and space! With a bore of 2 11/32 inches and 2 3/16 inches, respectively, it was apparently impossible (or too difficult) to fit four valve spring and cup assemblies of adequat size into the head. It is true that it took almost ten years for Miller to revert to the 4-valve layout, but then he did not design any new competition engines during that time! Early dirt track 4-cylinders were of the 2-valve layout simply because they were developments of a budget marine engine. True, that one was used in boat racing, so it was sort of a competition engine, but it was more or less a byproduct of a line of much larger, and non-competitive marine engines.