I think the player's whole team history is listed on the plaque. The hat worn on the plaque is subject to determination and I don't know which one they'd pick for that.
The plaque is to be representative of the player's career. So whatever hat they wore when playing is what goes on the plaque, irrespective of what cap the team is wearing at the time. So for example, Hank Aaron retired a Milwaukee Brewer, but the cap is from the 1960's Atlanta Braves. Larry Doby was elected in 1998, but his plaque is shown with a 1940's era Indians cap. The Houston Astros are the only team Biggio or Bagwell ever played for. Which Astros cap they use may be up for debate, but it has to be an Astros cap. There is no other option.
On a side note, I was looking at the HOF website the other day, which has players listed by their "primary team". Nolan Ryan's is listed as the Angels, even though he played the longest with the Astros and the Rangers cap is on his plaque.