HomePod mini is an AirPlay speaker. If your smart TV is able to connect to AirPlay devices for audio then it might work, but I don't think that is a common feature for third party manufacturers to implement. If it works at all it won't be dependent on whether or not you have an Apple Music subscription. You don't need an Apple Music subscription to play your purchased music on your iPad, and you can AirPlay music from your iPad via HomePod, purchased or not. HomePod should be able to play your non-hidden purchases directly, although for maximum flexibility, access to your playlists, and anything else in the Apple Music catalog, a subscription is recommended.
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