Thought I'd update my earlier posting. I tried locking my SIM as Clay142 suggested. Did not fix my problem but the iPad did seem to switch more readily to 3G when LTE got hung up.
I've upgraded to iOS 6.1.2 and for whatever reason I believe my problem is now fixed, sort of. The iPad works and LTE works when I take the iPad away from home. LTE does not work at the location where ROGERS, my ISP originally provisioned the device for the new nano SIM I tried as a remedy. LTE worked well right after the provision - I was on the phone with the ROGERS tech. But LTE stopped working about 4 hours later when I picked up the iPad again.
I think there are 2 things going on - Apple fixed something with iOS 6.1.2 because my iPad Mini is now working again when I take it away from home. It is still not working at home because ROGERS has very poor LTE coverage where my house is. The only reason I can think of why it worked during provisioning of the new nano SIM is that the ROGERS tech turned up the LTE coverage during the provisioning, then turned it back down afterwards. I have had this happen with 3G service at another town some years ago where ROGERS had very poor signal at that location - the ROGERS tech actually told me they would turn up the signal for me and they did. The signal did improve.
I drove around the neighbourhood and using the iPad I was able to locate the boundaries where the signal drops right off, from 4 bars to 2 bars. I cannot get a connection when LTE drops to 2 bars.
Finally, I am quite miffed at Apple Tech Support. Even though my iPad Mini was clearly under warranty and they acknowledged the problem, tech support simply stopped returning my calls and answering my emails even though the tech guy said he would get back to me. This happened immediately after I asked them to provide some means of obtaining a numeric readout for the cellular signal instead of the rather unreliable 1-2-3-4-5 bars indicator on the upper left corner. iPhone users can call a number and switch the bar display to a nurmeric display but since the iPad cannot dial, this feature doesn't work on the iPad.
When I discussed this need for a cellular signal strength indicator the tech guy seemed to indicate it requires the iPad to be jail-broken which is ridiculous from a functional standpoint because signal strength is already being indicated, albeit in bars. The tech guy seemed to "turn off" when I suggested that Apple look into providing such a feature.
After that, both tech guys I was working with simply disappeared. Very bizarre and unprofessional of Apple. I complained to Apple when I received a request to rate the tech support response I received. Hope they get their heads sorted out.