とする does not necessarily have multiple meanings. The translations may differ wildly, but the grammatical function is always the same, that being that it functions like a combination of the particles と and する.
If you have problems understanding とする there is probably a misunderstanding of either と or する. In this case it seems like it is と.
Disregarding english grammar and solely focusing on Japanese grammar, the sentence: レッスン消化としている most closely translates to:
“(It) Lessonlost-with acting.”
"(It)"here being the topic that was presented previously.
So when trying to keep in mind japanese grammar, it means that The Topic (は) outputs an action (する) simultaneously with (と) the presence of the noun (レッスン消化)
In more simple terms, A and B here is one unit, aka B acts-as (aka する) something that happens in direct correspondence-with (aka と) A.
This often translates to “Is considered” or “Acts as” and is simply a formal way of addressing a widely known fact.
The fact that けど is used with とする gives it a more contrastive nuance, like saying “Even though A is commonly considered, B”
In this case: “Same-day cancellations are technically treated as used lessons, but なんてなんて…”
If you just wanna know what とする means and dont like my emphasis on 1-to-1 grammar translations which may appear conveluded in english, here is a translation that works most of the time.
“Acting as/with (A), (B)”
However if you want to understand the nuances, you gotta go back to the 1-to-1 translation.