Plato, Phaedo, 63d--problem with relative pronoun

[63δ] κοινὸν γὰρ δὴ ἔμοιγε δοκεῖ καὶ ἡμῖν εἶναι ἀγαθὸν τοῦτο, καὶ ἅμα σοι ἡ ἀπολογία ἔσται, ἐὰν ἅπερ λέγεις ἡμᾶς πείσῃς.

I want to focus on this part:

ἐὰν ἅπερ λέγεις ἡμᾶς πείσῃ

After struggling with this for a long time, my trial translation is: “if [with, or by means of] what you say, you persuade us.”

I’m trying to justify this by LSJ, ὅς IV.2

  1. [select] the neut. of the Relat. is used in Att. to introduce a clause qualifying the whole of the principal clause which follows. . . .—so also the neut. pl. ἅ may mean with reference to that which

Have I got the grammar right?

Bailly: 7 avec double accus. πείθειν τινά τι, persuader qqe ch. à qqn, persuader ou convaincre qqn de qqe ch. HDT. 1, 163 ; ESCHL. Pr. 1063, etc. ; XÉN. Hier. 1, 16 ;
Liddell: with neut. pron., persuade one to or of a thing, τοῦτό γε οὐκ ἔπειθε τοὺς Φωκαιέας Hdt.1.163, cf. A.Pr.1064 (anap.), Pl.R.399b, etc.

Hope this helps.

Double accusative! I like your solution tico. Thanks.