"So while they fix that, anyone got suggestions for where we might find that mainframe backup so we can have R4 retrieve the communication logs?"

not certain if gather information , knowledge (galatic lore) , knowledge (Tactics) , perception or use computer - skill checks might be relevant to the subject of finding the backup, or having an idea of where it is BUT here goes checks for what it may be worth

gather information: (1d20+5)[9]
knowledge (galatic lore): (1d20+11)[24]
knowledge (Tactics): (1d20+6)[15]
perception (1d20+5)[11]
use computer: (1d20+6)[17]