Your other tool may still be or have been a combined pivot file and burnisher. It depends how it's been used in the past as to what it will be like now. One or both ends may be dressed as a burnisher - depending what sort of tool they presently appear to be, if one end is still a pivot file then you may wish to keep it as one.. a pivot file is useful so I'd keep it as one if it's in even partially worn condition.
Basically to dress a burnisher you could use traditional techniques such as oilstone dust on a lap with a bit of oil, or nowadays you could use a modern abrasive of a similar grade. The process creates tiny grooves in the tool from the abrasive which do a combination of really fine cutting and burnishing the pivots you apply the tool to. The burnisher would be moved across the abrasive and rolled over part way through the stroke, starting with a large face then rolled onto the next smaller face. You will see that there should be 2 sharp edges and 2 rounded edges, that will indicate which edges are rolled over and which shouldn't. It will probably take a bit of time if there are any imperfections or surface rust, etc to remove - you may choose to work on a side at a time to flatten and remove imperfections before dressing it.
New lanterns can occasionally be found but it is rare. More often these tools are sold with damage to the same parts. It is possible to make or repair the parts which are damaged with a combination of tools which will generally differ by whether they include a 90 degree double angle milling cutter or a tiny drill - for grooves or holes. Off the top of my head, the depth of grooves with a 90 degree cutter is a depth of 0.65 times the desired pivot diameter.
I'm assuming the issue is with the disk with tiny holes in it.. it can be possible to anneal the damaged end and to turn most of it away, leaving a boss concentric with the outer diameter which fits through the tool. The end of the boss can be tapped for a retaining screw for the new part, or the new part can be glued/ bonded on the end - both parts will want to be a nice close fit so that the new part remains concentric if drilled before being mounted to the old arbor / part of the tool. The new disk can be steel or brass (they don't see a massive amount of wear but may have abrasives embed in them, and will wear out quicker than steel). In order to drill one you'll need a way of indexing so that you can use the tools existing divisions (the star shaped part and guide). It is possible that a new disk may be able to be drilled or center drilled in place on the tool with some creativity.
I've also got a jocot tool with a damaged disk, shall repair it eventually but just use another for now.