Is luminous (pabhassa­ra­-citta) mind unconditioned and not impermanent?

Actually, no. Because pabhassa­ra­-citta must arise. Because if pabhassa­ra­-citta will never arise, it can't be defiled by incoming defilements which will arise to defile it, too.
Citta, viññāṇa, mano are the synonym of each other in SN Nidānavagga, assutavasutta:

“But, bhikkhus, as to that which is called ‘mind(citta)’ and ‘mentality(mano)’ and ‘consciousness(viññāṇa)’ —the uninstructed worldling is unable to experience revulsion towards it, unable to become dispassionate towards it and be liberated from it.

So, pabhassa­ra­-citta is conditioned (sankhara) and impermanent (anicca) by SN Khandhavagga, Aniccasutta:

consciousness is impermanent.

Therefore, paphassara-citta is included in one synonym of citta In abhidhamma, dhammasaṅgiṇī, kusalacittuppādakaṇṭa:

The citta which is arising while the occation (that 1st wholsome citta is arising) means citta, mano, mānasa, hadaya, paṇḍara (paphassara), mano, manāyatana, manindriya, viññāṇa, viññāṇa-khandha, and tajjā manoviññāṇadhātu — this is the citta which is arising while the occation.
(Translated version is not correct, so I did not quote from that. I have to translate myself.)