Hi!
What sort of cage is Pepper in or going to be in? How enclosed is it? What sort of heating set-up do you plan to have?
One very, very big issue I see with this is the heating. Especially during rain or colder weather and seasons. I would not recommend housing your hedgehog on your balcony, as sudden temperature shifts or prolonged exposure to lower temperatures can and will cause hibernation- which is often lethal if you cannot get to your hedgehog and warm him up before he falls asleep. The weather will also take a toll on the heating equipment itself, if you plan to set it up outside. Water can and will damage the electronics and could cause harm to Pepper.
Another issue is unmonitored substances getting into his cage. If it has a mesh or bar top, this will help, but will not stop things like pollen, dust, and smoke- these can and likely will cause respiratory infections, especially with long exposure times. One more thing that could be problematic is insects and bugs getting into his cage. While at first this may seem good- they eat insects in the wild, after all- it really is not. The bugs could carry any number of diseases that will harm your hedgie, from skin issues to internal issues. You also do not know what these bugs have eaten and, subsequently, what Pepper will be eating. If they have eaten something toxic, Pepper will then have eaten something toxic. With the list of food items that are toxic for these little guys, it's not a risk worth taking. Things like citrus fruits, potatoes, and bread are a few off the top of my head.
Pepper should be housed indoors, where it is far easier to keep an eye on him, his environment, and his temperatures (please, please make sure you have a thermostat and a heat lamp with a ceramic heat emitter- the thermostat will ensure his enclosure is kept at a set temperature). African pygmy hedgehogs need to be kept between 72F-80F or 24C-27C. They are also very prone to respiratory illnesses, mites, and other diseases.
As well, the more consistent and stable Pepper's environment is, the more comfortable and happy he will be.
Hope you and your little guy are doing well with the move!