Hi Clovis,
Thanks for putting the pictures up. That is unfortunate (what is the width?).
There are several things I can say. If the leather is very supple, don't worry about it, just use more pressure. The design would accommodate that, but only if the leather is very soft.
You could try running your hand up and down the edges, cupping them back over. But I think this would not be a permanent solution.
What would be the most likely long term solution is this: Take some tepid water with a drop of dish washing soap in it (one drop in 500 ml maybe). Wet a sponge and wring it out well so it is just damp. Quickly and uniformly run this over the surface of the strop. Then wet/wring the sponge and do it again after a minute. The leather should have been clearly dampened, but not wet. Wrap some plastic wrap around the whole strop, bending the edges over. But make sure there are no wrinkles in the plastic, marks made on the leather will not come out later. Let it sit over night.
Take the plastic off and lightly form the edges over as you would like them (if needed). A little extra at this stage wouldn't be bad. Do this several times over the next few hours.
Put a little neatsfoot oil on your hands and rub it well into the whole surface of the leather (just a little oil). If this disappears immediately, do it once more. This will change the draw somewhat, but probably for the better.
Personally, I think your leather may be dry. But just oil alone at this point (without the wet forming) most likely won't solve the cupping.
Good luck with it, padded strops are very nice and a good piece of kit. Having the middle length of a padded strop higher than the edges is no problem.
regards,
Torolf
ps. Don't think you need to baby the leather, make it do what you want it to, but gentle is good. My concerns about the amount of water have to do with the fact that you do not want to get the foam wet.