Interestingly enough, we could get away with the change if we used a different verb and added a comma:
- We will update the server to add more space.
- We will update the server, adding more space.
Both of those are grammatical. The first one explains the purpose for updating the server (to add more space). The second one explains a consequence of updating the server (adding more space).
The only problem with your S2 is that, in the phrase "including more space", the verb include doesn't fit well in that context. Grammatically, though, the verb include could work in different context. For example, a president of a university might say either one of these to the admissions office:
- We will accept more applicants to include more international students.
- We will accept more applicants, including international students.
The two sentences are not exactly equivalent, though. Once again, S1 states the driving reason for accepting more applicants (the university wants more international students enrolled). S2, on the other hand, clarifies the planned course of action: it's not just local student enrollments that will go up, but international students as well.