Rama discusses a similar situation in ShA OC 54:3ShA OC 54:3 where, after reciting Yishtabach, the congregation halted the prayer service for specific Mitzva/communal needs. He recommends in that case for the Chazzan to recite "some verses from Pesukei Dizimra" and say Kaddish "on them". (It seems to me that his specification of "from Pesukei Dizimra" is lav davka and any verses should suffice, though those may be thematically appropriate.)
Regarding your first point: I am not aware of an obligation on the Chazzan to fulfill illiterate congregants' obligation to say Yishtabach. Indeed it seems unlikely that there is one, as traditionally there is no mandated Chazzan for Yishtabach's sister-blessing of Baruch SheAmar (cf. ShA OC 53:1 who begins the Chazzan's role with Yishtabach). Furthermore, Rambam in his presentation of public prayer (Tefilla 9:1) omits the public recitation of Yishtabach entirely.