When you have a main verb of secondary tense, it is common to find a subordinate clause that was originally in subjunctive -
in the optative. And so:
..., αἰτεῖ, ἐπειδὰν αὐτοῖς οὕτω δοκῇ,... would become ...,ᾔτησεν, ἐπειδὴ αὐτοῖς οὕτω δοκοίη,...
ἐᾶν is infinitive of ἐάω, to let, suffer, allow, permit x (acc.) to do y (inf.). ἐᾶν is dependent on ᾔτησεν. (LSJ