Answer:

You are right when saying that breathing can become more difficult when activating your core. In some postures (e.g. plank, tree, hreadstand,warrior poses) it is extremely important to keep the core active so that the lumbar spine and the crossbones on your back are being protected.

In case you really have the feeling that you cannot breathe properly anymore, you are perhaps forcing the stabilisation too much, That happens often. In that case do only 50% of the pulling your belly and ribs in and up and breathe as deeply as you can.

Breathing is a very interesting subject in yoga. It is so that in Pranayama or breathing exercises and in Savasana, the end relaxation, you should breathe as deeply as possible into your lower belly.

Breathing into the belly calms you down physically, mentally and emotionally. 

When the yoga postures become more demanding or when you need strength in stabilising, you can change to breathing into the chest. The chest will then expand in every direction with the breathing.

Breathing in the chest will keep the strength in the core while the breathing does not flow in so deeply into the lower belly and it makes you asana yoga practice more stable.

There are some more dynamic yoga styles, such as Ashtanga Vinyasa yoga where only the chest breathing is used.