If the recess is deep enough, I would try attaching the finial with magnets.
Neodymium disc magnets are cheap, and even small ones can hold over 1Kg, which should be more than enough for a finial. You can get them with countersunk holes to attach with screws.
For this to work, the finial recess needs to be deep enough to hold a pair of disc magnets. I would try it this way, using magnets with countersunk screw holes:
Screw one magnet to the end of the pole. Make sure it's centred in diameter of the pole.
Screw the other magnet inside the recess of the finial. Be careful to centre this one too.
The magnets need to be in contact when assembled, so you will need to adjust the spacing by packing washers behind one or both magnets. Once you're close enough with the washers, you can then make micro-adjustments by backing out the screws slightly. For example, if you can get the magnets to within a few millimetres by packing with washers, then you can unscrew one of the screws by enough to let the magnets touch when assembled. So long as they're close enough to start with, they'll pull themselves together over a short distance.
That's not really a woodworking answer, but that's what I'd try first. The other options involve either reducing the diameter of the end of the pole, or widening the recess in the finial, and neither of those options are very easy, especially for a beginner. A slightly easier version might be to cut down the end of the pole to a square cross-section, then chisel out a corresponding square hole in the finial.