I would highly recommend that your networking team revisit their rules on address ranges.
If this is not an option, you can start by reading the AKS Egress documentation. It is mostly designed for public egress, but you will still be able to get useful information.
The ideal way to control the egress IP is to use a UDR to route all traffic to a NVA which will SNAT the traffic so that it all appears to come from a single IP. There are many solutions for this for public egress, but for private egress your options are to use Azure Firewall to SNAT a Private IP Range or use a 3rd party NVA that does something similar. The biggest drawback of the configuration is the cost associated with having an Azure Firewall added to your VNET.