I do know there are options with subscribing to Sirius, for example, and Sirius is cheap enough. The cost is low enough to consider it as an option.
Satellite is preferable - to me - than streaming from an already overtaxed wi-fi.
Since we have Metrocast, they have a thing called TV Everywhere. As far as I know, it will broadcast to your PC and to your phone. But I don't know of any other kind of device that can support it. I was hoping there was such a thing, but I'm not aware of it.
Sorry Sam, that was rude of me not getting back to respond to this yesterday.
I don't see where Metrocast's TV Everywhere has an app for streaming devices like Roku or Amazon's Fire TV Stick or Google's Chromecast, and I know that it doesn't have one for Apple TV. So it may be that you only have the smartphone and tablet (and iPod Touch) options for those music channels in particular. If I'm recalling correctly your son is fairly young? So you might not want to use a smartphone? You could get an old smartphone from someone that isn't using it anymore and just use it for that, it wouldn't need to have cellular service. And it could connect to any of a large variety of speakers either wirelessly or with a short wire coming out of the headphone jack or as a dock. Those kinds of speakers can be bought for maybe $20 and up, and even the cheap ones do a decent enough job for most people.
But there are lots of other music services that are free or fairly inexpensive. And some of them not only have smartphone and tablet apps but also work through the streaming devices I mentioned above, so they can just be plugged into a TV (e.g. with an HDMI cable). The last time I looked Sirius seemed pretty pricey by comparison.
It does have some advantages, but if all you need is music I suspect there are other better suited services. And some of them allow you to make music available offline so that you don't have to stream it all the time - for times when you don't have internet access or when streaming would use up limited data allowances or bandwidth.
You mention your WiFi already being taxed. You'd know your situation better than I would of course, but streaming music doesn't really use a lot of bandwidth. It's not nearly as bad as high quality video. And with some services you can adjust quality settings to reduce the bandwidth used.
Anyway, I'm still not sure enough about your situation to offer you less general feedback so all of that might be of little help.