Blow torch on a chip you want to save is probably not the best choice
Anyways, attaching wires to the pins is viable, but very time consuming and potentially difficult to diagnose a problem if one is attached oddly or something on the board interferes with reading it through the programmer. Do you have any logic analyzer test clips? You can get cheap variants off ebay and you'd probably need quite a few, but something with a little smt sized micro grabber on the end could do the trick, though attaching them all to the chip would take a couple minutes.
I think the test clip was the right idea, and yeah $50 for something to use on one project is a bit of a price.... but it's really the right tool for the job if you're trying to access them in circuit. If you just want the data on the chips, the best approach is to probably desolder and read from a socket, but again that requires equipment. If it's any help, you can get a decent cheap hot air station for around the same price as the clip.