You might need to do some re-evaluation of your base assumptions.

1. Andromeda is roughly the same size as the Milky Way. At least to the extent of our measurement margins of errors (ironically, it's easier for us to measure Andromeda's mass than the Milky Way's)
2. A typical satellite galaxy has a mass of about 10 million Solar Masses. Milky Way (and Andromeda) have about 1-2 trillion. To double the mass of the Milky Way, it would need to absorb about a 100k dwarf galaxies. There's only a couple of dozen.
3. Astronomers have mapped the nearby universe, and we know where the nearby galaxies are - There aren't enough to substantially change the next few billion years of galactic evolution in our neighborhood

As for your theory, well it's...different. Galaxies do merge, and the merger process does disrupt their shape - but we really go know how to model these mergers, and the description of the future of Andromeda and the Milky Way is pretty well established.

(Edit: Changed some of my points to better address OP's "theory")