Yeah, I think the network will probably not even stop you from broadcasting it because it's not a consensus rule, in fact it's not even a node policy. You can even make a 0 BTC output I think (provided you pay above the minimum dust fee as you have stated), and would still be valid even though it is technically worthless.
According to consensus rules, the total value of inputs must be equal or greater than the total value of outputs. This means even a transaction with zero fee and outputs worth zero can be valid. According to standard rules, the fee rate must be equal or greater than 1 sat/vbyte and the value of any of the outputs created in the transaction must be greater than the dust limit.
Almost all nodes follow these standard rules and reject any transaction creating an output worth less than the dust limit, regardless of the fee paid for the transaction.
Almost all nodes reject any transaction with the fee rate of less than 1 sat/vbyte, regardless of the value of outputs.
If you run your own node, you can broadcast a transaction with zero fee and an output with zero value, but other nodes will reject it and your transaction won't be propagated to the network. Such transaction must be sent directly to a mining pool.
