In your example "Using" is a participle — a form of verb having the qualities of both verbs and adjectives. It qualifies the 'key' with which he slit open the package and "Using... Package" is a nonrestrictive clause. We have intentionally substituted 'USING' by 'WITH' to show you that you have already answered your own question.
Essential/restrictive/defining/integrated or by wherever term we name such clauses, we cannot get rid of such clauses if we like to and they don't have commas before and after the sentence with relative pronoun with or without a preposition or participle.
As regards WHICH -vs- THAT, they substitute noun phrases in the source sentence and discharge the dual role of connecting the clause to a noun as well as becoming the structural part of the clause itself. Though the borderline between these two relative pronouns is defacing as to their uses, it is more or less agreed that a defining or non defining Clause can accept WHICH but THAT is restricted to defining clauses only. In written English we use a ' comma' before non restrictive clause as in your example; so that supports our use of WHICH here.
Again if it is a restrictive clause but preceded by a preposition, we always use ' which '.
- We have got some balls with which (X thatX)you can play.
Your example sentence may be viewed as an extension of an immediately preceding (pied piped) preposition which accepts only WH– forms.
What if we use THAT? 'That' in that event would become 'pointy that', a demonstrative pronoun losing its relative nature and the structure would necessitate an altogether different rendering perhaps like one hereinbelow:
To be precise, participle clauses are adjective clauses. Yours is an example of adjective clause nonrestrictive by nature. The relative pronoun WHICH in this construction undergoes internal movement within the pied piped participle.