There is something to be said for letting the IO system and OS handle the buffering and writes. All filehandles have a write bufer which only written when it is full. One way to reduce seek times is increase the buffer size for the opened files. The advantage is that Perl decides when the write the buffers, the OS decides when the write it to disk, and both are pretty good at this.
The big advantage of caching filehandles is that the open files can hold output in the buffers until they are full. If they are continually being closed and reopened, then each line is being written individually.
What is need is some way to keep a limited number of filehandles opened to keep from hitting the limit. A LRU cache would be perfect. It see a couple of modules that implement this. Or reimplementing it would be pretty easy.