fluffy had mastitis - it is important this gets treated asap. i would not check her upside down to be honest. just sit her on your lap with her facing away from you and have a look. if it's red, quite hard and/or swollen, it's probably mastitis. the duct will be blocked or getting there. there may be teeth marks as the babies are still trying to get the milk - this would be really painful. this needs veterinary assessment.
the treatment may require antibiotics - obviously not ideal given she's nursing. warm compresses on the area and massage. i managed to get fluffy better with warm compresses and massage several times a day (every few hours). she wasn't pleased, i was super scratched. again, this needs a vet to have a look at it (take babies with mum, make sure they're all warm) - it may not be possible to treat her with warm compress and massage alone (i was given two days to see if it gets any better before putting her on antibiotics and it cleared).
in any case, you have to pick her up, not just to examine her but to treat as well. when they're sick, it matters not whether they like being picked up or not, you have to do it. mae, my rescue girl actually bit me twice that hard i bled (she is the only chinchilla who's ever done that) in the first week i had her but i had to pick her up to keep her alive. she absolutely loves me now. they may not understand it, but you do
if mastitis is untreated, she WILL stop nursing, she will run away from the kits, she may lose the ability to produce milk on the side where the teat with mastitis is and the inflammation/infection may spread which is potentially fatal.
i suggest you get her to the vet asap. if she stops nursing, you're in for weeks of hand feeding. i had triplets i needed to keep alive for three weeks before she resumed her duties - this was hand feeding every two hours day and night and each feed took over half an hour. it wasn't fun. go while she's still nursing!