Combinatorial optimization - is there a formal name for this problem?
I don't know the name of your problem, if there is one.

But, concerning the complexity: The decision-version of your problem seems to be strongly NP-complete.
You can easily reduce ExactCoverBy3Sets to it:

Given a ground set $X = \{1, \ldots, 3q\}$ and a set $\mathcal{C}$ of 3-element subsets $C_i$ of $X$. Is there a subcollection $\mathcal{C'} \subseteq \mathcal{C}$ of disjoint subsets that cover $X$ ?

Assume we have $3q$ services $1,\ldots,3q$, each with a price of $q+1$. Furthermore, for each $C_i$, we have a package of the corresponding three elements at a price of $1$. Now, given the list $1,\ldots,3q$ of services to order, there is a solution with cost at most $q$ if and only if there is a solution to the given instance of ExactCoverBy3Sets. Since there are no double orders and since  "You can only use a package if you can completely fill it with services", the sets are disjoint and cover the whole ground set $X$.

So, you won't succeed in finding a polynomial (exact) algorithm or an FPTAS. However, you can search for approximation algorithms for the above problem and/or the related problems SetPacking and MinimumCover (cf. Garey/Johnson, "Computers and Intractibility", Problems SP2, SP3 and SP5).