Thanks for taking the time to write this up, I appreciate the explanation.The principle is the same with exogenous testosterone. Instead of internal HPTA regulation of free testosterone there is external regulation via our control of the dose rate. Your example would get complicated if you're talking about injecting pure testosterone and DHT. But typically these hormones are injected as long-acting esters, which lead to a slow and steady introduction of the base hormones into circulation. The DHT does effectively make less SHBG available to bind testosterone. I illustrated above that this does not affect free testosterone when T and DHT are being introduced at constant rates.
Another point about your example is that overall metabolism of testosterone is limited by the rate of production or dosing. If testosterone is slowly being absorbed from a depot then you cannot on average metabolize testosterone at a faster rate than it is coming in. If you provide a sudden spike of pure DHT then you would have a brief surge in free T and its metabolism. But as the DHT declines things go the other way, with a short-lived excess of available SHBG causing free T to fall below baseline. If you take the average of free testosterone over the transient period then you'd find that it's the same as during the original steady state.
Somebody proposed a sponge analogy for SHBG, which is fleshed-out here. I find it useful for getting a sense of SHBG's role as a storage buffer for testosterone. It also highlights that the main action we care about is the "flow" of free testosterone from production/dosing to eventual metabolism and elimination.