The Tur (OC 46) had the same issue as you. He writes that the custom around him was to say Birkot haTorah and some Torah content before Pesukei deZimra but after reciting the "Leolam Yehei" and "Attah Hu" prayers which themselves contain Torah content including verses. Accordingly he changed his own practice to say Birkot haTorah immediately after Elohai Neshama at the very beginning of the day.
However, other Rishonim (such as Agur (92)) do not have a problem with this set up because the verses are being recited דרך תחנונים by way of supplication, not דרך לימוד by way of learning. Indeed, by the fact that the Tur thought to change it, we see longstanding custom was in accordance with this view. The Gra records in his commentary to the below-cited Shulchan Aruch that in all the very old Siddurim or records of the order of prayer he could find, he saw that all of them place Birkot haTorah later.
The Shuchan Aruch (OC 46:9) rules:
לא יקרא פסוקים קודם ברכת התורה, אעפ"י שהוא אומרם דרך תחנונים. ויש אומרים שאין לחוש, כיון שאינו אומרם אלא דרך תחנונים, ונכון לחוש לסברא ראשונה. הגה: אבל המנהג כסברה אחרונה שהרי בימי הסליחות מתפללים הסליחות ואחר כך מברכין על התורה עם סדר שאר הברכות וכן בכל יום כשנכנסין לבית הכנסת אומרים כמה פסוקים ותחנונים ואחר כך מברכין על התורה
One should not read verses before reciting Birkot haTorah, even though they are being said by way of supplication. And some say one should not worry about this because they are only said by way of supplication. And it is proper to worry about the first opinion. Gloss: But the custom is like the latter opinion, for on the days of Selichot we pray the Selichot and afterwards bless on the Torah with the rest of the blessings, and so too every day when we enter the synagague we say various verses and supplications and then bless on the Torah.
So it seems there is strong traditional basis to not worry about saying verses by way of supplication before reciting Birkot haTorah.