You may have already worked through many of the things I will say, but others may also benefit from hearing this.
The first thing I would recommend is take the camera and Teradek to a location with a bright room and one that has a cable (not DSL) internet connection of at least 10mbps upload. Do a test using CBR (constant bitrate) at 2mbps. Test going to the church webcasting platform and also to Youtube. You may learn a lot from this exercise.
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Doing a speedtest closely simulates a webcast done with CBR-constant bitrate, where a constant stream of same sized chunks of data are sent to the destination. Speedtests do not simulate a webcast done with VBR-variable bitrate.
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Another factor is latency. If latency varies a lot, it can cause all sorts of problems with a video stream. The internet needs to be "consistent" in sending your stream of data. Go to the Azure website and test your latency to a specific datacenter.
https://www.azurespeed.com/Azure/Latency
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If WIFI is on, you don't really know how much upload bandwidth you have dedicated to the Teradek - try turning it off and see if picture improves.
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Using variable, "adaptive" bitrate will try to maintain a certain picture "quality" at the expense of bandwith. Bitrate can and will vary below AND FAR ABOVE the target bitrate you set. This is a dangerous choice when you have limited bandwidth at the Chapel. Constant bitrate will provide a more reliable connection at the expense of picture quality (only during motion).
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Something that can affect the Teradek's ability to encode a picture efficiently is the amount of noise in the picture. If the lighting is on the low side (which is true in a lot of chapels) there will be a lot more noise in the many small blocks of the picture. The encoder will see this simply as a constantly changing block and "change" and will encode it for every single frame it sends out. The encoder will have to do this continually for each and every small block of noise. If you are transmitting VBR (variable) this will raise you overall transmitted bitrate and leave less bandwidth for the "real" parts of the picture - the worst part is when motion occurs - the encoder bitrate will rise to whatever it needs to encode the motion+noise. Consequently, it is more likely you will exceed the chapels available bandwidth. You don't EVER want to exceed the available bandwidth or even get close to it.
If you are transmitting CBR(constant) and you have noise, it will always leave less bandwidth for the "real" parts of the changing picture - but you never exceed the bitrate you set, and keep you within the chapels available bandwidth.
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You can minimize picture noise that the encoder has to process by setting the camera aperture wide open or increasing the light, then using a shutter speed to get the exposure right. At the same time, you must be aware that using a really slow shutter speed will also cause motion blur and the encoder will again have to work harder.
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Finally, be cautious about thinking the webcast statistics show you the real peak bitrates. It may be doing short term averages, or just reporting a snapshot bitrate periodically to you- neither of which is a true peak.