The recurring problem however is that the printed sheets have a thin unprinted, blank border...This 2mm border keeps appearing in my prints.
This is subject to the capabilities of your printer. There's nothing you can do in Illustrator to change the hardware capabilities of your printer.
This is for silkscreening, so even though its only a 2mm border, this gap prevents me from putting all the printed sheets together as one design.
Are you actually planning on abutting edges of printed pages in order to make a single positive for exposing a single silkscreen?
Even if your printer did print all the way off of the edge of its paper, you'd still have to deal with the exposure light that is going to shine between the abutting sheets.
If you don't see that as a problem, then why don't you just trim the 2mm printable area off the sheets?
Hope this is making sense?
Unfortunately my printer, Canon iX6540, can only print up to A3. So for anything larger I would need to print my design over multiple pages.
"you'd still have to deal with the exposure light that is going to shine between the abutting sheets."
Fair enough, makes sense. So what would you recommend?
"If you don't see that as a problem, then why don't you just trim the 2mm printable area off the sheets?"
Maybe a border of 1-2cm on each page so that they can overlap, ensuring better design registration? Problem with that however is that the border cuts into the design. So 1cm or 2mm of the design ends up being missing.
What would you recommend? I've been told that Coreldraw is the solution, but unfortunately I don't have this software nor the money to purchase. There has to be a solution in Illustrator / Indesign?
If you have InDesign, place your Illustrator file into an InDesign file, and print it tiled from InDesign.
InDesign does its tiles with overlap you are talking about, and with internal registration marks to help with lining up the sheet. I've often wished that Illustrator had the same system; I often have to place Illustrator-created dielines into InDesign just to get a useful tiled output.