I came to this solution (after trying many of the suggestions posted) by commenting out parts of the script to see what kind of effect it had.
For example, I have a couple of subs that I invoke after the uploaded file is written to disk that do various things depending on the file type. The subs require a couple of different packages that I was use'ing at the top of the script. If I commented out those subs and their required packages, I was able to get larger files uploaded with no problem. The next thing was to figure why these subs were causing the script to be killed before they were even invoked.
The reason I didn't think they were a problem initially was because any file that was 35 KB or under worked flawlessly, while anything above caused the script to be killed before it could write the file to disk, hence they were never called. By changing the packages that these subs need from being use'd to require'd, I was able to upload much larger files without the script being killed and still have them work their magic.