Even at non relativistic speeds, accelerating bodies increase in mass, but I agree with you, the difference is negligible.
I think we are looking at this from two different angles. The postulate of special relativity says that there is no experiment you can perform to indicate that you are moving if you are inside a constant velocity reference frame.
Your use of the 'constant velocity' term seems wrong to me. I believe his example was an elevator in free fall that was accelerating because of the Gravitational field, an accelerating body is not in a 'constant velocity' situation. In particular Einstein stated that you couldn't tell the difference from the inside of a box sitting on the earth or a box in space accelerating at 32 feet per second squared.
Essentially what Einstein was saying was that you are accelerating at 32 feet per second squared as you are reading this post.
The elevator in free fall is General Relativity.
The airplane moving at constant “uniform motion” is Special Relativity.
Both examples are correct.