Piyyutim for Shabbos Rosh Chodesh Av with a Bris

I'm a little confused what piyyutim are recited according to Minhag Frankfurt on Shabbos Rosh Chodesh Av with a bris. The link below seems to say the Elokeychem for Shabbos Rosh Chodesh is recited, but then says that the Elokeychem for Shabbos with a bris is recited. Are they both recited? If so, what is the procedure
www.moreshetashkenaz.com/life2_3.htm

Melech

There seems to be a very good reason to say the Elokeychem of a Bris only on Shabbos Rosh Chodesh Av: in order that we honor the bris in the davening through some piyyut. Although Yomim Tovim have so much חשיבות that their piyyutim take over and preempt those of a Bris from being said (apparently so as not to take away from the Yom Tov, so that not even the ofan of a bris is recited on a day of Yom Tov with no Ofan) the same thing doesn't seem to be true for regular שבתות, where it appears that a conscious effort is made to divide the piyyutim in such a way that each event that is מחיב a piyyut is honored with one. So although it was the zulas of bris that was normally chosen as the piyyut to honor the bris on a Shabbos with other piyyutim, it would seem that it makes sense to say the Elokeychem instead where require saying the zulas of bein hametzorim in order to properly mark that time.

Melech