As mentioned, the program is most likely printing directly to the printer hardware port, instead of "properly" by using the Windows API.
You wouldn't redirect the LPT1 to LPT3, you would instead redirect the LPT1 to your network share.
net use LPT1: \\pcname\sharedprinter /persistent:yes
(as mentioned above). NOW, if your software "behaves", and prints using the printer BIOS calls, this may intercept it, and route it over to your network printer. If it doesn't print, you can try one other trick.
Go into your printer control panel, and add your dot matrix printer drivers to LPT1, just as if you were going to connect the printer directly. THEN, turn on Printer Pooling between LPT1 and your shared network printer. If you're lucky, your DOS program will try to print to LPT1, but Windows will say "Oh, LPT1 isn't available.... but it's pooled with this printer over here...." and route your print job to your network printer.
Just my 2¢
"In order to start solving a problem, one must first identify its owner." --Me
--Greg