Skipping or spinning...

[unfazed]

Is the needle bearing actually worn? I have only ever replaced one, but many roller bearings. Last roller bearing I replaced was on Downeys bike and the needle and sprocket hub bearings were fine.

Best way to check is is with the axle in and the new bearing lightly seated, unfortunately you have driven the new bearing all the way in and I do not think it is possible to remove the needle bearing without removing the roller bearing first. Thankfully for you if the needle bearing is worn, new roller bearing are very cheap even for branded ones, last FAG one I bought from a local supplier was €10.
Worst case scenario is you will have to  remove the roller bearing again and use it to push out the seal thus keeping the seal intact.

Last time I removed one of these needle bearing I used an off cut of steel 300mm long from the local engineering shop the same diameter as the spacer. Dry heat and a good few thumps of a Club hammer should get it out, just ensure it is well supported at a height than the length of the bearing and the space for the bearing to drop out is clear.. It will much harder to get out than the roller.