I haven't found any references to Thomas yet, I'm assuming that the RC orphanage is probably not an option as you mention in your Kilkenny post (see:
Need help finding Betts) that he was an episcopal minister, and the records you've found relating to the family are from CofI churches.
I did find some Bett's in the teaching profession in Slater's directory of 1846... since they share both a profession and relatively uncommon surname with Thomas it seems possible that they are connected..
Slaters 1846
Geo. Betts, master of Irishtown charity school, Irishtown (Dublin)
Sarah Betts , mistress of infants' school, Irishtown (Dublin)
Mr. Betts, master of parochial school, Clara (Co. Offaly)
Shane