Learn English – Differences between “of a degree that” and “to such a degree that”

Those are largely similar; however, from here and here:

• in order to and so as to are more formal and used before verbs like know, seem, understand: ... I asked in order to know....
• so that is less formal - and often reduced to so in informal use. It also is used with modal verbs like can: ... so that I can ...
• so that and in order to need a subject/verb: ... I did this so that I could...

Google Ngrams gives, for American English (red = so that, blue = in order to, orange = so as to):

and for British English:

Doesn't look like there is much of a difference between American and British English in that matter.

In this context, I would say that you could use either word, but they mean different things.

Let's start with:

I copy of a lot of files from A to C through B.

This means that, to copy a file from A to C, the files must pass through B (perhaps because A and C are not directly connected to each other). That would align with Meanings Nos. 7 & 8 in the Collins Learner's Dictionary (CLD):

through (prep.)
7 To go through a system means to move around it or to pass from one end of it to the other.
8 If something goes into an object and comes out of the other side, you can say that it passes through the object.

However, if someone were to say:

I copy of a lot of files from A to C by B.

I would interpret that to mean:

I copy of a lot of files from A to C by using B.

As CLD Meaning #3 says:

by (prep.)
3 If you do something by a particular means, you do it using that thing.

I think this wording might be better if B does some kind of work on the file (such as scanning for viruses) before copying the files onto C.

However, if only English were that cut and dried! There's also this meaning of by:

by (prep.)
11 If a person or vehicle goes by you, they move past you without stopping.

If by was being used in that sense of the word, that sounds fairly synonymous with through, so there may not be any additional processing going on after all.

As a native speaker, my advice would be to use through if the files are just being copied, and by using if there is some other processing going on, but that wouldn't be the ONLY correct way to use these two prepositions. Most prepositions are extremely versatile words, and then can often be used in overlapping contexts.