Well, that’s unfortunate. If you don’t find a new AVR with an additional audio out you might have to split the HDMI signal.
Here is one thing (below). I’m not sure the device outputs the full HDMI signal. Unclear. It probably does but not assuming that. If it is the full signal then you could come out of the AVR with the HDMI cable, put this splitter in to get the audio, and continue to the TV with the HDMI out of the splitter.
If it doesn’t pass the whole HDMI signal through to the TV you would have to split the HDMI signal above that device, and use the above device just to extract the audio signal. Go to the TV with the other leg of the splitter below.
So worst case you would have AVR, HDMI out with full video/audio, splitter to get two hdmi signals, take one of those two to the TV for video, take the other of the two and connect to the first device above, to convert the sound to a format the TV Transmitter can handle.
Will this do something to the timing of the audio? I wouldn’t be surprised. Most new AVRs have an adjustment to line up the sound with the lips.
Going to the TV and getting the sound off the toslink may work just as well, but this gives you another option.