According to the user manual, the Blackburn rack is required. The seat snaps in to the rack, so the old Schwinn rack won't work.
It's almost certainly cheaper and easier to just buy the correct rack, as well as safer. The risk of wrecking the seat is IMO too high.
You almost certainly can't attach that seat to that rack while keeping your warranty intact or maintaining any way to transfer legal liability away from yourself.
If you think the rack is strong enough to hold the seat and child, and it physically fits over the rack, it would be relatively straightforward to mount. Drill mounting holes to allow you to drop dome or flat head bolts through, then put metal strips under the rack to accept those bolts and hold the seat onto the rack. There are a lot of judgement calls to make during this process, but I have seen it done (and been asked to repair a seat held on like this that had obviously been used a lot). Some baskets are help on like this, but obviously they're not designed to carry babies.
Whether this is safe depends on a whole lot of things. Safety is relative, and compared to just balancing the child on the rack this is very safe indeed. But compared to using the proper rack.. not very safe. And possibly not a lot cheaper - you could end up spending as much on bolts, steel strips and tape/padding as the proper rack costs.