OK Softbank should work. It should work for your Apple Music streaming and all other LTE enhanced features with the limitations on the incoming and outgoing cellular calls.
First a disclaimer is required, while I have not actually tested this process it should work for you using Softbank. Before each Phase below, please power off both the iPhone and the Watch when swapping your wife's nano-SIM between iPhones.
Phase 1: As you guessed you should pair the Watch with your wife's (Phone Number) nano-SIM but it must installed in your physical iPhone. Once the pairing is complete and you have tested the LTE functions then you can move on to Phase 2.
Note: Do not un-pair the watch unless your wife's nano-SIM is installed in your iPhone.
Phase 2: Put your nano-SIM back in your iPhone. As a reminder: your Watch will still be paired to your wife's iPhone / Carrier and carrier phone numbers (iPhone & Watch). Your Watch's LTE coverage is not your Carrier's coverage so just keep that in mind there may be LTE coverage differences between your iPhone and the Watch.
Note: In addition to streaming Apple Music. You will also be able to make cellular calls, they will just appear to come from your wife's phone number. You can also answer calls for your wife's phone number or even be annoyed by those incoming calls since there is no way keep the watch from accepting incoming calls.
Dave