Every day I go to work, the implied threat is that my company could assign all my work to someone cheaper--generally speaking, give 1/Nth of my work to each of my N co-workers, then advertise an open position if that puts the team over capacity--and abruptly end our work-for-pay relationship. Or maybe they allow a tester to jump up into development, and give them tester+1 pay instead of developer pay. If I meet with my manager with competing offer in hand, that may invert the status quo of the power balance for all of 15 minutes. How unnerving for them.

You always need the company more than the company needs you. That's a corollary to the whole point of having a company, which is to produce an entity separate from and greater than its constituent parts.

Personally, I'd just keep silent about the amount of the competing offer and ask for a raise. They can do their own calculations about how much they can afford to give, and if they can't match the competing offer without knowing what it is, I'd take it. Telling them what that offer is makes as much sense as divulging your salary history when seeking out competing offers, which is to say it does not make sense. Let companies do their own calculation regarding how much you are worth to them, instead of copying the answers off of each other.