1. "We remain yours truly, Dombey and Sons" makes sense.
2. "We remain, best regards, Dombey and Sons" is nonsense. A mess.
These salutations have meaning, even though they are formulaic.
Sentence one indicates that the Dombeys remain true (loyal, honest) servants/suppliers/clients to the company.
Your second example is better, but could be improved. It doesn't stick out as being obviously wrong as the first one does.
"I assure you of our best attention at all times, (signed) sincerely (by) <signature>
There's a wide range of possible endings to a letter, so it shouldn't be difficult to use one that is both polite, formal, traditional, and that also makes sense. But you can't just mix and match with anything.