If I understand your question correctly, what you're getting at is what if the US didn't use a decimal currency? So the US would use something like the old British system of 12 pence to the shilling, 20 shilling to the pound?
If that's what you mean to ask, then the question is how would the US subdivide its currency. Historically the US used Spanish dollars far more than pounds. The modern USD is originally based on the Spanish dollar and short of completely altering the early American economy, I can't see why the US would ever use the pound as its basis over the Spanish dollar. However, historically the US has had a fondness for dividing the Spanish dollar up into bits, that is 1/8 of a dollar or 12.5 cents. I suspect that if the US must use a non-decimal system it would be based on this system. The weirdness of having a coin worth 12.5 cents could be resolved by doubling 12.5 to 25 - so you get 25 cents to the bit and 8 bits to the dollar; 200 cents to the dollar all told. Hopefully this would be imperial enough for your purposes.
Now supposing the U.S. used this "bit-dollar" the real question is what would change? Before the wave of British and former Dominion decimalizations in the 60s and early 70s, I suspect no one would seriously care. The strength of the US economy would still ensure that the dollar was the default currency for the world. After say 1990, computers are sufficiently advanced that the effects of having a weird currency subdivision wouldn't impact high finance at all. That leaves an interesting window between 1970 and 1990 when you could get some real butterfly effects. For instance as the Japanese economy booms in the 80s you may get some of the other Pacific Rim countries favoring the Yen over the weird-to-count Dollar as their reserve currency at least moreso than OTL. When the bottom falls out of the Japanese economy this could trigger the 1997 Asian Financial Crisis a couple of years early, or have it happen on time but worse. The problem is once you get to this point there are a lot of moving parts and its very hard to figure out what the effects could be - you could be looking at an accelerated Dot-com Bubble, but you could also be looking at a lessened one.
There's also all the small behavioral stuff: people throughout US history might spend their money slightly differently because of the psychological effects of different currency subdivision. Admittedly this might not be huge, cognitive biases like "first number bias" wouldn't be impacted much by a change in the trailing numbers of a price. Pre-printed menus and such would probably be written in a weird format like $$.xyz where xyz < 200 cents.
Hopefully someone else has some ideas - I'm really just spitballing now.
