My AT&T mobile hotspot isn't compatible to support the internet connection on my Xbox One. So I'm wondering if I get a mobile hotspot booster, will it let me use my mobile hotspot with the Xbox?
No. If your mobile data connection is not allowing you to use Xbox Live, it will be because of your phone or provider's limitations, not because of the signal strength. If it was a signal strength issue a better solution would be to plug your phone into the console directly and use it like that.
It's worth noting that even if you could use your mobile data for your Xbox, this would be a bad idea. Updates will use a large amount of mobile data and that's assuming you don't try to download any games (which can be 10s of GB in size). This would leave you with either no mobile data left, or a very large unexpected phone bill.