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BUT, they would both have absorbed roughly the same amount of energy.

They really wouldn't, because we have already shown that the bullet will have more kinetic energy than the rifle will...
We also know that from the workenergy relationships that work done is equal to the change in energies. Work is also equal to force over distance (or the time rate of change of momentum). The target downrange will adsorb all the energy over a shorter distance than the shooter, and because the rate of momentum change is much higher, will see a higher force number than the shooter...
Think of it this way... If you have two objects of the same momentum, and one applies a force over a short distance (the bullet) and one applies a force over a longer distance (the rifle against your shoulder), the work done must be equal in each case. To compensate, the force for the shorter distance has to be higher than the longer distance.
And remember, its time rate of change of momentum... The rate of change is higher for a bullet than it is a rifle...
Now, the problem is that this all uses particle dynamic maths, which is not an accurate representation of what happens when a bullet hits a body. That would require a continuum dynamic model to truly show what kind of force number you'd see...