So, after googling godaddy 1024 file limit and reading a few results, it seems like your best option might be to find a new host.
Barring that, the least painful way I can think of to fix the problem would be to logically organize the files into subdirectories and then use Apache's rewrite rules to map the old paths to the new paths. This would save you changing anything in the actual application. The following would be one example:
1) Download the directory to your local computer;
2) Organize the files into sub-directories, it shouldn't be too bad with only 7000 files and the 1024 limit. For example, you could create 36 sub directories, one for each letter of the alphabet and one for each number, 0-9, and then place each subset of files into the appropriate subdirectory: file awhatever.gif would go into a/, file 11whatever would go into 1/, etc.
3) Create / edit the .htaccess file and create a rewrite rule that would map any request for product_images/awhatever.gif to product_images/a/awhatever.gif.
The .htaccess file would look something like this:
RewriteEngine On RewriteRule ^(.)([^/]+)$ $1/$1$2 [L]