Here's interesting Google search results.
Ai-ai kasa no nure tsubame 7hits
Aiai kasa no nure tsubame 0hit
Aiaikasa no nure tsubame 0hit
Ai-ai gasa no nure tsubame 0hit
Aiai gasa no nure tsubame 0hit
Aiaigasa no nure tsubame 1hit
相合傘の濡れ燕 1hit(this page in JREF)
I checked all other possible forms 相合傘の濡燕, 相合傘の濡れつばめ, 相合傘のぬれつばめ,,,,,あいあいがさのぬれつばめ on Google, but there is NO site where this "proverb" is written in Japanese. What does this result suggest?
Interestingly, the only site that has the correct reading "aiaigasa" says that "Aiaigasa no nure tsubame" is the same meaning as 一石二鳥 "Kill two birds with one stone".
http://www.lit.osaka-cu.ac.jp/UCRC/2006/ja/issue/pdf/pdf_zasshi06/08arimi.pdf
(pdf file)
FYI, I found several misreadings similar to "aiaigasa" in the book you mentioned...