Use of a possessive determiner or "the"

There is first a question concerning the role of the participial clause "consisting of letters". Not using a comma after "one" adds a restriction on the collection; it is not another collection besides the collection of paintings but a new collection of letters and that way, you do not have two collections but three. This means also that there is in the context two collections of letters and that the locutor is considering the other one of these two by opposition to one that should have a reference in the context. This makes for a background context of more complexity, rather puzzling, but not an impossible one nor one that is too complex: the locutor is concerned at the same time with telling what he /she will do about a collection of books and with a rectification to the effect that what is involved is not a supposed collection of letters but another collection, also a collection of letters. The example context below fits this case.

• — They had some suggestions about your collection of letters from ten years ago, how it was sold.
  — I will sell his collection of books in the same way I did with my collection of paintings and the other one consisting of letters.

It follows that a comma is necessary so as to make the clause a descriptive one, thereby eliciting "the other one" or "my other one" as meaning "other than the collection of paintings", and corroborating the fact that there are just two collections involved from the point of view of the locutor.

I will sell his collection of books in the same way I did with my collection of paintings and the other one, consisting of letters.

Whether to use "the" or "my" is just a matter of context, a matter of what the locutor wants to communicate; if another collection has been mentioned as being the locutor's or if somehow the persons being talked to know that there is one more collection and that it is the locutor's, then "the" and "my" convey the same information. If the persons talked to do not know of another collection, then "the" tells them that there exists another collection but nothing as to whose collection it is. If they know of another collection, "the" tells them that the collection that is being mentioned is that one they know of, but they do not necessarily learn that it is the locutor's. If on the other hand "my" is used it follows additionally that that collection is the locutor's property (or a collection with which he/she is closely associated), but that does not tell them in case they do not know whose collection of letters it was prior to the communication that the collection in question (referred to as "my other collection") is the same.

I will sell his collection of books in the same way I did with my collection of paintings and the other one consisting of letters.

As I read that, I have no idea to whom the collection of letters belongs. It could be his, yours, or someone else's.

I will assume that it is yours:

"I will sell his collection of books in the same way I did with my collection of paintings and the other one consisting of my collection of letters."

In reality, as there will be context and so, "I will sell his collection of books in the same way I did my collections of paintings and letters" could work.