Post a picture of the stopper. I've never seen one that was thick right in the middle, though some have legs on the sides. The middle section is normally thin. I use the same short needle size all the time with no trouble across many injections.
Just starting out with these and got some from a different source. The stopper sits lower than I'm used to, so my usual short needle needs a lot of force to push through and bends half the time. Thinking about ordering longer ones or moving the liquid into some spare empty vials I have. Any problems with doing the transfer?