I guess I don't quite understand what your issue is. You can receive faxes in fax only mode, but when someone sends a fax while it is in audio mode it won't work?
Assuming that is your issue, it probably will never work. The gateway identifies a call coming in as data or voice. If it is voice, it will try to determine if it is a FAX call can use either 1) FAX relay or 2) FAX pass through depending on how the gateway is configured. The ATA 186 will, under most circumstances, negotiate different codecs depending on the type of call. So if the call coming in is a voice call, it won't renegotiate when you start sending a FAX.
Keep in mind the ATA 186 is a cheap device. It is designed to work with a single analog device. That being said, it really isn't quited for what you are looking to do. The proper device for this is the VG200 series or an IAD device.
So what can you do? Depends on your gateway.
If you have fax relay enabled, see if fax pass through can be used. What kind of codec is being used right now? It may be possible to add some dial peers to force a G711 call (which may or may not pass the FAX). If your gateway supports analog VWICs, I would suggest putting one in and using that to connect to this FAX machine. You probably won't have the issues associated with the Codec since it won't leave the device. You may end up having to get a POTS line to handle this device. It is a fairly uncommon device on VoIP networks.
Last but not least, call Cisco.
It is what it is!!
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A+, Net+, I-Net+, Certified Web Master, MCP, MCSA, MCSE, CCNA, CCDA, and few others (I got bored one day)