Ukraine gave up nukes in a separate agreement in 1994, under guarantees from Russia and USA and UK that Ukraine's territorial integrity won't be violated and in case of such violation would be protected. Now that Russia blatantly violated that agreement, does it mean that Ukraine can claim breach of the Budapest Memorandum assignment and stake a claim on UNSC seat or other parts of USSR package?
Has the agreement been violated? The Wikipedia article appears to provide a positive answer, with multiple sources that are much more authoritative than whatever you could get on this site. Russia denies it of course. It's in the nature of international law that there isn't necessarily any way to adjudicate such disputes, even when the answer might seem obvious to you or me.
Can it claim something? A country can always claim anything but what would this achieve? The memorandum itself certainly does not lay out any procedure to seek redress (some international agreements do) or even merely hint at what Ukraine could ask in return. Given Ukraine's current predicament, there is no reason why anybody would take any outlandish claim seriously.
One thing seems obvious: A permanent security council seat for Ukraine does not pass the laugh test. The composition of the security council is such a huge can of worms that it seems very hard to see how it could change, let alone to benefit Ukraine!
Formally, the addition of a new permanent member would require amending the UN charter and could therefore be vetoed by any of the current permanent members (it's in the logic of the system: If they did not have such strong guarantees, the big powers would not have gotten on board in the first place). You can bet that Russia would stop that one.
I would think a permanent security council seat for Ukraine wasn't remotely plausible even when the country did have nuclear weapons on its territory. They were going to go and Ukraine wasn't going to get much in return. A vague agreement with weak guarantees (“commitment to seek immediate United Nations Security Council action to provide assistance to Ukraine […] if Ukraine should become a victim of an act of aggression or an object of a threat of aggression in which nuclear weapons are used” which implies by contrast that the parties do not even commit themselves to referring the matter to the UNSC if the threat does not involve nuclear weapons, let alone do anything else) was all there was in the cards. Now the weapons are gone and Ukraine really hasn't got any leverage on anybody.