The basic rule of borer[1] requires that three conditions be fulfilled in order to permissibly separate an object that is in a pile from others: 1) One takes the desired object (ochel) from the undesired objects (pesolet). 2) One plans to use the desired object soon. 3) One does not use a utensil made especially for selecting in the process. Your case seems to fail #2. However, it is possible that there is no problem, as we will explain, and there are also relatively simple ways to obviate the possible problem.
Rav Ovadya Yosef[2] justifies the practice of those who, in order to expedite the setting of the table on Shabbat morning, remove forks, knives, and spoons from a pile of silverware and arrange them in groups. In classic form, he does so by offering several possible reasons why the procedure may be permitted, even though each reason is not sufficiently convincing in its own right.
The Aruch HaShulchan[3] wonders how it is permitted to pick out one type of silverware from a bunch of assorted ones. His first suggestion is that because of their large size, each utensil is distinct in the eyes of the one who selects. Thus, the process of removing what he wants is not considered borer but simply taking. In fact, the Terumat HaDeshen[4] already posited this idea but was reluctant to rely upon it without further indications of leniency. However, several Acharonim question this proposition.[5] The matter may depend on how tightly packed the silverware is.
The Pri Megadim[6] raises the following important point. The Rambam[7] rules that if one has two useful types of items before him, he may take out the one he wants to use soon. The time factor can be understood in two ways. Perhaps the fact that one will be used before the other turns the former into ochel and the latter into relative pesolet. As mentioned above, the rule is that we may take ochel from pesolet, but not vice versa. If that is the issue, reasons the Pri Megadim, then in a case where one will use both groups of items at the same time in the future, there would be no problem because there is no distinction between ochel and pesolet. The Bi’ur Halacha[8] is convinced that this is not the explanation of the time factor. Rather, there is a special dispensation for one who takes an object to use in the short term, as this is considered “in the manner of consumption,” and it is, therefore, permitted. In contrast, selection for later use is more like classical borer.
There are additional, weaker grounds for leniency. One is the possibility that the Torah prohibition of borer applies only to things which grow from the ground. Another is the claim that preparations for the next meal are considered short-term even if the meal is significantly later.
We note that, based on the aforementioned reasons and others, many are of the custom to allow separating the silverware unconditionally. However, we will share suggestions to conform to the majority opinion among poskim, who are not lenient.[9] One is to keep the silverware in a pile and set the table soon before the meal begins, as indicated by the Rambam. Please note that it is not sufficient to do the separation soon before setting the table, if that is long before the meal commences.[10] Another possibility is to not select utensils from the pile but to use the opportunity when the utensils are in your hand to create separate piles for each category, as follows. After taking them one by one indiscriminately in order to wash or dry, put each one in a separate pile by type. With a little organization, this system need not waste too much time. Others suggest artificially undoing the mixture by throwing it across a surface (like pick-up sticks) and then selecting as desired.[11]