You need to separate the male kit because, yes he could impregnate the mother. There are plenty of instances of dads/sons and mother/daughters getting along fine, so it's definitely worth trying with supervision. There is a chance any one of these chins could up and decide they hate one another and turn on each other. They may not. This is the risk you take by choosing to house more than one and why you need more than one cage at all times in case this happens. You won't know until you attempt a proper introduction. If you do a search there are some tips on here in numerous threads about how to do it properly.
Just because you post a question on here does not mean we are going to ignore the train wreck that this situation is. You also need to separate the parents or you are going to get more kits, possibility of more hand feeding, possibility of a csection, possibility of a dead chin or kits, possibility of more inbred chins. Just because you had one litter go well does not mean the next one will. I had a female who has given birth several times before have stillborn mummified kits that she couldnt pass and needed and emergency csection and spay. She still ended up dying several hundred dollars later. If your gf isn't willing to pay for a new cage then how or is she going to pay for that? Does the animal just sit there and suffer?
Most responsible breeders would give a female who just had triplets at least a short break before putting her back into a pregnancy situation. I mean think about it - I highly doubt any woman who just had a baby would be ready to be pregnant immediately. The female's body needs time to recover and heal to carry a pregnancy. By putting them together you are risking that her body might not be fully prepared to carry and deliver healthy babies = complications.
It sounds like you are in control of them at the moment, so you do have some choice. If she wants to keep breeding tell her she needs to buy another cage which is true anyways and you could keep everyone separate and try to reason with her. If she balks then you will clearly see her commitment to doing this appropriately.
And I am sorry you feel like this isn't your job, but you both have an obligation and responsibility to make sure the kits and animals are taken care of and this includes buying proper supplies (i.e. baby safe cages) and housing them appropriately while in either of your care. A 3ft tall cage is not made to house a breeding pair period, much less young kits. It's a safety hazard for the pregnant mother and babies. As I said before, failure to plan for the scenarios you are creating is no excuse to keep perpetuating more babies from a pair that may or may not be related and may or may not be carrying genetic issues. I mean you wouldn't breed shelter/mutt dogs repeatedly because you didn't want to put forth the effort to keep them separately, would you? Why should chinchillas be any different? I don't know maybe you would, but maybe I just don't understand the logic here as I certainly wouldn't.
IMO time for one or both of you to step up and do the right thing.