Salamun Alaikum,
In my humble opinion, the best way to understand the term 'ayah' in 45:6 is in light of both the grammatical construction of the verse and the surrounding context. You are correct to interpret 'ayah' in 45:3-5 as 'signs / portents'
However, in 45:6, how are the signs being conveyed to the listener? Of course, they are being related in the form of communications and a book which is confirmed in 45:2. "The revelation of the Book is from Allah, the Mighty, the Wise".
So of course the reference in 45:6 is to the 'signs' related in 45:3, 45:4 and 45:5 but in 45:6 these are being summarised and referred to in the form of a recitation / rehearsal and read to the listener as part of a wider Quran. This is clear from the Arabic "natlluha alayka' (We recite them to you) where the verb 'yatlu' has been clearly expressed as an action of recitation / rehearsal.
Therefore, the nominative noun 'ayat' is linked to the pronoun 'ha' of 'tatluha' which in turn is linked to the imperfect verb 'natlu' (recite / rehearsed). So in this context, the ayah, are being 'recited' in the form of communications / verses which links back to 45:2.
This is further supported by the Arabic "fabi'ayyi hadithin ba'dallahi waayatihi" (Then in what statement (Hadith) after God and His Verses). Therefore, 'ayathihi' is linked with 'hadithin' (statement) as a form of communication and best interpreted as 'verses' in my humble opinion.
It would be difficult to read it as "Then in what communication (Hadith) will they believe after God and His 'signs' (ayatihi)?". Here the word 'Hadith' falls out of kilter with the interpretation of ayatihi as 'signs' and makes little sense.
Finally, as you will appreciate 3:7 should not be taken lightly. Here 'ayatun' clearly refers to the verses of the Quran as best interpretation. This is the same for 2:106, 16:101 or 11:1
011:001
"Alif Lam Ra (This is) a Book, whose verses (ayatuhu) are made decisive, then are they made plain, from the Wise, All-aware"
016:101 (Part)
"When We substitute one verse (ayatan) for another..."
This is why the word 'ayat' is nuanced and has always been well appreciated as such by classical Arabic lexicographers and Arabic speakers.
I hope that helps, God willing.
Joseph.