Canonical Transformation in Quantum Phase Space

I am looking for a unitary representation $hat T$ of the following canonical transformation

begin{align}
q_1&rightarrow q_2 &p_1&rightarrow p_2
q_2 &rightarrow -q_1&p_2&rightarrow -p_1
end{align}

which is a 90°-rotation in the $(q_1,q_2)$-subspace of a 4-dim phase-space. It is therefore a point-transformation, since it does not mix positions and momenta. $hat T$ acts as

$$
hat T hat q_1 hat T^dagger =hat q_2 quad hat T hat p_1 hat T^dagger =hat p_2
hat T hat q_2 hat T^dagger =-hat q_1 quad hat T hat p_2 hat T^dagger =-hat p_2
$$

One guess of mine is

$$
hat T = e^{-i( p_1(q_1-q_2)-p_2(q_2+q_1))}
$$

but I do not know a way of proofing it, apart from expanding the exponentials and then computing everything brute-force, e.g. in

$$
bigg( sum_n^infty frac{i^j( p_1(q_1-q_2)-p_2(q_2+q_1))^n}{n!}bigg) hat q_1 bigg( sum_m^infty frac{i^m( p_1(q_1-q_2)-p_2(q_2+q_1))^m}{m!}bigg) = hat q_2.
$$

One would have to commute $hat q_1$ to the left, which seems ridiculously laborious to me. Is there an easy way to find $hat T$ for such a point-transformation? And if one must resort to guessing, is there an easy way to proof that what one has found acts in the right way?

I am deeply grateful for any help!