The perfect subjunctive doesn't make sense to me here, coming after a present conditional verb. It feels like it should be either arriviamo or arriveremmo. Is that simply not how Italian does things?
I think it would be more correct to say "cosa sarebbe successo se fossimo arrivati ultimi" OR "cosa succederebbe se arrivassimo ultimi".
Yes, I think that the sentence is grammatically uncorrect.
Your version is the right one