In a previous life I used to do this kind of thing, when I worked for the water board, before we sent the cars back to the lease company.
Is the bumper an awkward colour?
We used to use the plastic caps that go over screw holes in flat pack furniture, file them down to fit and then epoxy them in from the back. If the car had any black plastic trim (even if it was just wing mirrors) then we'd leave them black. Otherwise we'd paint them roughly the right colour and we only once got picked up on it after the car got sold on and the owner had thought it was the covers for the rear parking sensors. (we must have done about 10 a week for 3 years)
I reckon that'd be a pretty safe bet.
Once we did have an Audi A6 to do which took a bit more thinking, I plastic welded some plasticard from the rear, built up a layer of fine grade milliput from the front and then painted it with a touch up stick. Then Polished the paint around it to blend it in nicely. Cost about £15 in materials but it was pretty flawless
What kind of car is it? can you get anything to roughly match?
Alex