I can't seem to find a reasonable way to get to get past this part. I've carefully jumped back across the bridge, but the Kleer Skeletons just keep building up. Once there are enough of them, the bridge extends, allowing them to reach me. How can I defeat this large mob without having to take advantage of an exploit?
I played through this on Serious mode, and I got past this section by rocket-jumping up onto one of the the buildings and laying waste to all the enemies from there. They could still throw stuff at me, but for the most part they were nullified. I don't know if that's what you're calling an exploit, though. Serious Sam harkens back to an older age of gaming where this sort of stuff wasn't really considered an exploit. In fact, in the non-HD version, a lot of secrets were accessible only by rocket jumping.
The other thing to do is make sure you get the secret on this level, since it contains ammo which will be useful. If I remember right, there's actually a cannon in that secret ammo depot, which can help a bit in this fight since the Kleers tend to line up nicely. Since I barely remember the details of it, I'll just cut and paste what I wrote in my Secrets FAQ for the game: