Just to answer my own question, in case anyone else with the same challenge finds it.
After a lot of trial, error and looking at other bridge configurations on the internet, I found out that I needed to turn off notifications. The following bridge configuration will allow you to bridge messages from a local mosquitto server to thingspeak 🙂
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connection thingspeak
address mqtt.thingspeak.com
notifications false
cleansession true
topic # out 0 remote/thingspeak/ channels/
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Happy MQTT'ing