Congrats on your new chinchillas. That would be neat if you could breed but I would try to avoid it with those two unless they both come from reputable breeders and were sold with breeding in mind. Typically pet chinchillas are not kept for breeding because they are inferior quality or may come from a line that has congenital defects that can be heartbreaking to pet owners if they manifest in the kits.
If you've established both are good breeding quality and you wish to breed, certainly do the research, and make sure you have the appropriate cages (1/2" x 1/2" bar spacing maximum, low ceiling, no shelving, etc.) for when it is closer to time or your kits will escape and die, fall and get injured and die (both which I have observed on the board) or starve. Since you have experience raising small animals I doubt you'll be prone to as many n00b mistakes as I've witnessed here. Another suggestion would be to have the female examined by a vet to make sure she is suitable for breeding. Sometimes you'll get one with not enough pelvic separation and be at risk for CPD related issues. Also calcium supplements are usually recommended for pregnant and nursing chinchillas.
As far as separation goes I'd say the best thing you can to is try it and see what happens. Some chinchillas will do just fine and others will go absolutely haywire. It is going to be stressful but you'll just have to weigh them and keep close track of their food consumption each day. If they go anorexic it's gone too far. (Chinchillas should not be allowed to stop eating for more than 24 hours or they can develop irreversible GI stasis.) If they just turned a year I'm hopeful they'll be able to separate. Chinchillas should not be bred until after 1 year. Older chinchillas are going to be more set in their ways. Another option is to try to get them to pair with another chinchilla of the same sex.
Neutering is risky and very invasive. The gonads are not the protruding parts you can see but are deeper inside. I've helped others on this board with nursing chins back to health after surgery. I've observed the procedure is generally survivable but not always and usually intense care and syringe feeding is required to get them to recover. It must be done by an expert veterinarian and you may have to travel to find one. It should be a last resort IMO.