Thanks for pointing to the info on H&B site. Things have changed quite a bit on their site since I last looked.
I can understand why H&B may feel it necessary to point out the potential issue of increased stomach acid and levothyroxine absorption, but in reality I would be surprised if it actually makes any significant difference. The idea of taking betaine HCl with meals, is to directly increase stomach acid level at the time of food intake. However, we should not be taking levothyroxine anywhere near mealtimes. So I would think that your stomach acid level will be in it's 'normal for you' state when you take your levothyroxine, regardless of taking the betaine HCL.
Obviously though, if you were to become aware of changes in symptoms that suggested the possibility of your thyroxine dose being too high, then a visit to GP to request a check on your thyroid hormone blood level would make sense
Regarding the dose of betaine HCL, have a look here under the heading 'Treatment of hypochlorhydria'