Here is what I am seeing. When I view source and look at the link for Floppy Straw Hats I see the URL
http://www.surfdome.com/baku_hats_-_baku_congo_hat_-_volcano-108584?i
and this link shows me a 200 when I run through it directly. This is probably what Screaming Frog is doing. I would re-run the frog and set the user agent to Google Bot just to see what happens there.
Now when I view that link in the browser and I hover over it and right click the URL and copy I get
http://go.redirectingat.com/?id=207X327&xs=2&url=http%3A%2F%2Fsurfdome.com%2F&xguid=e3d44c6f62047018ab665c4fe5392990&xword=surfdome&xcreo=300001&xpid=875260&sref=http%3A%2F%2Fwww.myfashionlife.com%2Farchives%2F2013%2F07%2F16%2Fget-anne-hathaways-paper-denim-and-cloth-look%2F&pref=
When you run this, you get the 302 redirect to the target page
If you scroll down to the bottom, you see the skimlinks JavaScript that is doing this manipulation. FYI it is also adding redirect a link to "Surfdome" at the end of that same line. This is not linked at all in the source code. You have a simple JS rewirte action going on there.
So the bot sees the regular URL and the human sees a redirect via JavaScript.
Depending on if you wear a white or grey hat, this could be considered "cloaking"
https://support.google.com/webmasters/answer/66355?hl=en
"Cloaking refers to the practice of presenting different content or URLs to human users and search engines."
This is not the traditional use of the redirect. You would often see a completely different page shown to the bot vs the human using JS versus your example of just showing a 2 different links on a page. That said, Google is reading more and more JS these days http://googlewebmastercentral.blogspot.com/2011/11/get-post-and-safely-surfacing-more-of.html
Your issue is not about the 302 passing link equity, but if you want to get penalized for cloaking or not.
The other point that comes up is that since you are paying these bloggers to have this link on the site, I would call these paid links
https://support.google.com/webmasters/answer/66356?hl=en
I know you said, "these are organic links" we are now just paying for the referrals. Well, if Google finds that this is worth penalizing, then you have no one to argue with but yourself.
As I see it, you have 2 choices
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No follow the links and do not use them for link juice but to pay for traffic.
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Lose the redirects and use the links for ranking benefit (plus you can still get some traffic).
Honestly, seems like you were already getting organic links and traffic for free, so not sure why you would pay people for what you already had. I am sure this helped to get additional links, but you just need to consider the points above to see what is more important for your site.
Hope this helps!