遣ってみようか - Why is there a か?

My textbook has this exchange:


Why is there a か in 遣ってみようか?

Wouldn't the usual, formal response to a ませんか invitation be ましょう...?

ましょうか (or volitional form + か for casual) is itself an invitation or offer, and not a response to an invitation, correct?

Is this similar to かs used in rhetorical expressions, like そうか?

Thanks!

～（よ）うか can be used for an agreement/acceptance to an invitation or offer.
e.g.
A: そろそろ休憩にしない？
B: うん、そうしようか。

C: ぜひ、議長をやってもらえませんか。
D: わかった。そこまで言うならやってみようか。

か usually add a less-positive nuance comparing to just ～（よ）う, or it could be a humble expression, avoiding too pushy.

Excellent; thanks very much!