Turns out the dance hall was a Friar's Club, & a notorious one at that, involving lots of booze, gambling, & police raids.1 Quite a large structure, too. For those not familiar w/ the area, that would be the modern-day Milwaukie waterfront park behind the island to the left.
Milwaukie residents steered clear of the hall of ill repute, but Portlanders came by the streetcar load, & then were ferried to the island - hence, presumably, the concrete landing on the west shore.2
I couldn't figure out the duration of the club's raucous operation. Grain export magnate Peter Kerr acquired the island in 1910, & Oregon went dry on January 1, 1916, & I doubt the establishment survived prohibition as a juice bar or somesuch.
Kerr donated the island to the city of Portland in 1940.3
As an unexpected bonus, it turns out the Milwaukie Museum has in its possession the sole surviving mule-drawn streetcar of the Portland Street Railway!
Established in 1872 by rail baron Ben Holladay, the PSR ran its diminutive streetcars up & down Front Street from Glisan to Caruthers & back. When Holladay died in 1887 ownership transferred to his brother Joe, who kept the line running the length of its entire franchise, which expired September 9, 1896, earning it the distinction of Portland's 1st & last horse-drawn streetcar company.4
Should you be so inclined, the Milwaukie Museum is located at 3737 S.E. Adams St. in Milwaukie, but is only open 11-3 Saturday + Sundays. Haven't been myself, but it seems a great small museum!
1 House of History: Inside the Milwuakie Museum
3 Brochure, Friends of Elk Rock Island, Page 2
4 Labbe, John. Fares, Please! Those Portland Trolley Years. Caxton Printers, Ltd. Caldwell, Idaho. (1980) p. 23-26