There's not an easy way to differentiate the two 1890 Giants teams and two 1890 A's teams in the search engine, so you'll have to use baseball-reference to parse out which players play on each team.
A bit late since only the two Athletic teams are left, but I found that if I use the search settings:
Team/League Div: American Association (Players League or National League depending on which one you're looking for)
Min Games at: All 0
Season(s): 1890
Order By: Team
Then By: Salary (decreasing)
Max Matches: 150
You can find the team you're looking for but it might be on page 2. Seems a lot easier than going back and forth between bbr.