Communicating over a private port with another vlan is identical to using a non-private port. The difference is in communicating with other ports in the SAME vlan. Traffic for secure ports is not flooded within the vlan.
I think you might have a configuration issue, or your switch may not have L3 capabilities. The model you specified should be able to switch on Layer3.
Did you configure the vlan in the vtp database?
Have you configured an interface vlan x for every vlan you use?
It may be something else, feel free to post your config so we can help you out.
We are pleased to announce availability of Beta software for 16.6.3. 16.6.3 will be the second rebuild on the 16.6 release train targeted towards Catalyst 9500/9400/9300/3850/3650 switching platforms. We are looking for early feedback from custome...