I don't have a spare but I think it'd be fairly easy to fabricate one from piano wire.
You have a reasonably good template in one of your photos (we'll ignore the break).
I haven't looked lately but it used to be hobby shops carried piano wire in stock.
I had a string controlled, .049 gas driven model aircraft as a youngster. Dad got pretty good at fabricating landing gear for it.
I think a large socket would make a really good form for the circular part. Due to the spring action of the wire something probably considerably smaller than the finished size (You may need to overlap the ends on the socket to get the desired final size.).
If you successfully make one please post your procedure (including the socket size and wire diameter) to help the next guy.
HTH and TIA
Lou
The link below is what the last spring I ordered looked like (Poulan weedeater (OrderTree) - NLA). It has overall length and diameter dimensions.
http://www.mediafire.com/file/dohsde8811e80hv/FarmSawTakeupSpoolSpring.pdf/file
My IPL part number is 432129 (add prefix 507 for Poulan weed eater).
That updated and became 507429572.
Edited 07 Nov 2018 13:30 PDT
I don't recall where I got the following number 507429892 - 507429892 is the black piece that engages with the saw (flywheel) and actually turns/starts your FarmSaw, you probably do not need this part.
I dug it up in one of these old posts by me
https://www.arboristsite.com/community/threads/pioneer-chainsaws.70698/page-320
Damn... memory skipped a cog... again!
GL