the only way I’ve ever been able to gain a significant salary increase is by lying

I've never been given a significant salary increase or promotion without making up some bullshit job offer. I've only been called on it once by an owner who suspected it was a competitor. I was able to produce a fake offer letter to prove otherwise. If you keep getting the run around, just make some shit up. Now, if you know you don't have a leg to stand on or work for real sticklers, maybe don't.