After a year or so of meetings where my manager agreed that I was underpaid and he was "working" on an increase that never came, I bluffed the competing offer. Did my best do so without outright lying, using terms like "I was exploring other options", and establishing a deadline for their response. They never did, but if the probing questions had gone too deep, I would have cited confidentiality. It was a bit sporty, but I knew other firms were hiring, and it worked. Had it not, I was only two years away from FIRE, and I would have just sucked up the embarrassment of not leaving. There was no way I was going to jump ship, learn a whole new system, just to quit.
There was absolutely no awkwardness afterwards, and if anything, perhaps a little more respect. It probably happens more often than we think.
As another poster mentioned, lines such as, "I really don't want to leave, but I just can't pass up such a promotion, etc" are very effective. Especially if you really are a good employee. In the event they do not offer to match, you can actually leave (if you have a real offer;), or talk about how great your current company is vs the offering company. There are dozens of ways to do this gracefully without any hard feelings or risking much, even if it doesn't work out
I worked with a guy who negotiated with offers several times over his career. One just has to be careful not to over do it, which could get you get a reputation for it, but even then, in a short Mustachian career, how likely is that? Plus, Megacorp's tendency to restructure every few months, makes them vulnerable to competitors when it comes to retaining their employees, they know this, take advantage of it when you can. Looking back, I probably should have done it at least one or two more times.
As a side note, I would have taken a manager mentioning getting a competing offer as an invitation to get one. Any chance they said this with a wink and a nod?