The proposal - to require advance shareholder approval for large acquisitions - received about 10 percent of votes in favour, the company confirmed to Reuters, casting doubt on the group's ability to gather enough support to scupper the Shire deal.
The Shire deal would turn Takeda into one of the world's largest drugmakers but the group says it would require taking on too much financial risk.
The 130 member-group formed by ex-Takeda employees told Reuters earlier this month it did not expect its proposal to pass on Thursday.
It said it is working to persuade a third of shareholders to vote against a company proposal at a Takeda shareholder meeting later this year or early next year to approve an issue of new stock to help fund the Shire deal, making it a de facto vote on the deal itself.
Thursday's proposal reflected concerns among some investors about the attempt by Chief Executive Christophe Weber - the company's first foreign head - to clinch the largest-ever overseas purchase by a Japanese company.
The move would turn Takeda into one of the world's most indebted drugmakers, though Weber has insisted he can deliver savings to make the deal a big winner for shareholders.
Shares in Takeda and Shire both gained more than 2 percent following the AGM.
Takeda's shares are nevertheless down 18 percent at 4,524 yen since the company first said at the end of March it was considering bidding for Shire.
With uncertainty over whether the acquisition will be approved by shareholders and given concern over the size of the deal, "investors are taking a wait-and-see stance" towards Takeda stock, UBS analyst Atsushi Seki wrote in a note to clients last week.
In the note, UBS upgraded its recommendation on Takeda stock to "buy" from "neutral" and raised its price target to 6,700 yen. It also said a survey of 60 Japanese investors found 31 percent viewed the Shire deal negatively.
Proxy advisor Institutional Shareholder Services (ISS) recommended voting against Thursday's proposal, saying that while its "proponents raise legitimate concerns about the planned acquisition of Shire" there was an "absence of apparent grounds to cast doubt over the board's objectivity or competence".
(Reporting by Sam Nussey and Ritsuko Shimizu; Additional reporting by Ben Hirschler; Editing by Christopher Cushing)
By Sam Nussey