Some inverters are simply designed that way. I have seen this in most cases especially in Hybrid inverters where the inverter is designed not to give output while on inverter mode.
Can you let me know the model of your inverter? Or better still you can consult the manual of your inverter. Just make sure the inverter is inverting and giving out right output on normal operation. After this I can help you diagnose it to see if this is a malfunction or simply the design.
If this is the design, then you may want to install a relay to the inverter switch or an external switch for a manual changeover without turning on/off the inverter.