Learn English – What does “remain current on” mean

As you stated, the "scarlet letter" was the letter A that Hester Prynne was made to wear in The Scarlet Letter to signify that she was an adulterer. In the book, she is treated as a social pariah (outcast) because of the letter. Most people would not associate or interact with her unless they had to, and when they did it was with a look of disgust.

In the sentence you quoted, the implied meaning is that Republican senators who proclaimed themselves to be a part of the Tea Party are now trying to figure out if it will help them or if it caused them to become outcasts with regard to their fellow Republicans as well as citizens in general.

I imagine it's an echo of this expression (as defined by OED)...

(strictly) for the birds
trivial, worthless; appealing only to gullible people. orig. and chiefly U.S. colloq.

Dobbs clearly isn't interested in anything Howard has to say. He's implying that despite Howard's "evangelical" enthusiasm, nobody else would be interested either, except gullible people as above.

OP is on the right track with bird = fellow - but here it's specifically birdbrained (stupid) fellows. The word catch is also used figuratively here (Dobbs means interest, draw in, captivate, impress).