A few (hopefully helpful) observations...
While all the keys on a horn will wear over a period of time, you tend to find less end-wear on keys that are mounted on point/pivot screws. It does happen, but far less rapidly than on keys mounted on rod/hinge screws.
It's also far less likely to happen on keys that get less use - such as the bell keys.
If I see significant axial (end-to-end) play on bell keys the first thing I'm going to suspect is that the pillars have been knocked out of alignment. This is extremely common; if the horn takes a knock, the resultant shock might not leave a dent...but it can cause the large keys to act like slide hammers - and this will push the pillars apart.
The solution is to reposition the pillars to remove the axial play...and this is typically done by carefully tapping them back into position.
Grinding material off a removable compound bell key pillar is a risky thing to do. If you don't maintain a flat mating surface the pillar will not be held securely against the mount...and if the screw that holds it in place has a limited depth of travel you may not be able to achieve a suitable level of torque. It also knackers the value of the horn.
Fitting washers to barrels is a popular 'bodge' - but it's more commonly used on keys mounted on rod screws.
There's a good reason for this.
Examine the photo below:
Here's a typical point screw mounted key.
The principal behind its action is that the key barrel is supported on the tapered point of the screw. Ideally there should be full contact between the surface of the point and the corresponding hole in the key barrel. There should be the tiniest amount of clearance between the end of the key barrel and the face of the pillar.
In this diagram you can see that there's clearance between the screw point and the hole in the barrel, and between the end of the barrel and the pillar. It's a worn key, in other words - and it's probably a good representation of the keywork on your tenor.
You can probably see that if you moved the pillar in a little, it would take up the play around the point of the screw and reduce the gap between the pillar and the key barrel. Chances are it would solve the problem.
What you've done is solve half the problem - and the least important half. You've popped a washer on the end of the barrel to take up the gap...but the point screw may well be floating around in free space (for some of the time, at least).
So - your key barrel no longer exhibits axial play....but if you grip the barrel ends and give them a wiggle there's a good chance that they'll move...because your fix has done nothing to reduce radial (side-to-side) play.
However....there may be some reasonably good news.
If your horn has parallel point screws (with a cylindrical head rather than a tapered one) it doesn't matter quite so much where the screw hole is in relation to the screw's point. As long as the screw is able to hold the key in place,, it'll work (though it may not be quite as tight as it was before...and because it's a parallel point it'll never be that tight anyway).
If your keys have no radial play - leave them alone. If they do...remove the keys and very gently tap the pillars towards each other. It's best to tap both ends so that you maintain the correct angle of the screw - and after tapping the removable compound pillar you should always check the tightness of the screw that holds it on the mount.
