[tyc85]

I think the short answer is no, currently there isn't a reasonably priced LTE routers that you can deployed by yourself. There are products available though. That's why the question can be interesting to be figured out to a certain extent.

Whether device look for carrier in unlicensed spectrum: the answer is yes, Licensed Assisted Access is happening, though slowly, as with any things related to carrier. See this link: https://support.t-mobile.com/thread/144981

Finally, it's certainly not something you can only do with SDR, but then doing it with SDR could be what makes it appealing though, it the price can be somewhat brought down. Good Wi-Fi routers nowadays cost a lot anyways, and some even wants you to pay monthly fee, no?