AT&T cell booster work with just hotspot

@themorgans66  No. And even if you could get it to work (unlikely), that connection type would not be supported if issues occurred. Without a wired internet connection, your only other option would be to purchase a cellular booster (the AT&T Cell Booster is a femtocell, not a cellular booster) if you can get at least one and preferably two bars of cellular signal. See my Cellular Booster Guide, the link is the second one in my sig line. That is a good primer on what cellular boosters can and can not do.

^^ What OttoPylot said.

The AT&T Cell Booster, regardless of its name, is a Femtocell which needs an Internet connection for the voice data it sends and receives over the Internet to complete your mobile phone connections.  If you had a good, reliable cellular connection to allow your hotspot to function, why would you need the Femtocell?!  Your cell phones could use that good, reliable cellular connection directly.

Thank you both. Do you have any recommendations on what device would help in our particular situation? 

Read my Cellular Booster Guide as I mentioned. That will tell you if a cellular booster will work for you or not. As @JefferMC said, if you have a decent cellular connection without internet then a Cell Booster won't work and your choices are down to a cellular booster, which may give you a bit better cellular reception.

As far as internet goes, have you looked into satellite? Something like StarLink?