[This version has been updated in response to comments from the OP]
Recall that $B$ gives an endofunctor of the category of abelian topological groups. We can apply $B$ to the obvious map $\mathbb{Z}\to\mathbb{Z}/2$ to get a homomorphism $S^1=B\mathbb{Z}\to B\mathbb{Z}/2$, then compose with $\det\colon U(2)\to S^1$ to get a homomorphism $U(2)\to B\mathbb{Z}/2$. The fibre of this is the group $\{(g,u)\in U(2)\times S^1:\det(g)=u^2\}$, which is easily identified with $S^3\times S^1$. We can now apply $B$ to get a fibration
$$ B\mathbb{Z}/2 \to BS^3\times BS^1 \to BU(2) \to K(\mathbb{Z}/2,2). $$
(Note here that $S^1$ is implicitly embedded in $U(2)$ as the centre, rather than the top left copy of $U(1)$.) We can now apply the functor $\text{Map}_*(\mathbb{C}P^2,-)$ to get a fibration
$$ 0 \to \text{Map}_*(\mathbb{C}P^2,BS^3)\times \mathbb{Z} \to \text{Map}_*(\mathbb{C}P^2,BU(2)) \to \mathbb{Z}/2. $$
Now let $P_k$ denote the subspace of $\text{Map}_*(\mathbb{C}P^2,BU(2))$ consisting of maps that have degree $k$ on the bottom cell, and put $M=\text{Map}_*(\mathbb{C}P^2,BS^3)$. From the above fibration we can see that $P_k\simeq M$ whenever $k$ is even.
Next, we have a cofibre sequence $S^3\xrightarrow{\eta}S^2\to\mathbb{C}P^2$ (where $\eta$ is the Hopf map), giving a fibration
$$ M \to \Omega^2BS^3 = \Omega S^3 \xrightarrow{\eta^*} \Omega^3BS^3=\Omega^2S^3. $$
It is known that $\Omega^2S^3=S^1\times W$ for a space $W$ whose homotopy and (co)homology groups are all finite. It is easy to see that $\text{Map}_*(\Omega S^3,S^1)$ is contractible, so $\eta^*$ really lands in $W$.
The above fibration gives short exact sequences $C_{i+1}\to\pi_iM\to K_i$, where $K_i$ and $C_i$ are the kernel and cokernel of the map $u\mapsto u\circ\Sigma^{i-1}\eta$ from $\pi_i\Omega S^3=\pi_{i+1}S^3$ to $\pi_{i+2}S^3$. At least the $2$-torsion part of these groups can be read off (in a substantial range) from Toda's book "Composition methods in the homotopy groups of spheres". If I have got everything straight, the first few nontrivial groups are as follows:
- $K_2$ is $\mathbb{Z}$, generated by $2\iota$.
- $C_4$ is $\mathbb{Z}/2$, generated by (the image of) $\nu'$. Here $\nu'\colon S^6=S^3\wedge S^3\to S^3$ can be obtained from the commutator map $S^3\times S^3\to S^3$ by collapsing out the axes; it is conceivable that this description could be illuminating.
- $K_5$ is $\mathbb{Z}/2$, generated by $2\nu'$
- $K_7$ is $\mathbb{Z}/2$, generated by $\nu'\eta^2$
- $C_{11}$ is $\mathbb{Z}/2$, generated by $\epsilon$
This gives $\pi_2M=\mathbb{Z}$ and $\pi_3M=\pi_5M=\pi_7M=\pi_{10}M=\mathbb{Z}/2$ (ignoring any odd torsion).
You can probably also extract some information about the mod $2$ (co)homology of $M$ using the same fibration. Recall that $H_*(\Omega S^3;\mathbb{Z})=\mathbb{Z}[x]$ with $|x|=2$, but the multiplication here comes from the loop structure and $\eta^*$ is not a loop map, so it will not give a ring map in homology. Instead we should use $H^*(\Omega S^3;\mathbb{Z}/2)$, which is generated by classes $u_i$ in degree $2^{i+1}$ (for $i\geq 0$) satisfying $u_i^2=0$. It is known that $\Omega S^3$ splits stably as a wedge of spheres, and this means that all Steenrod operations are trivial. On the other hand, $H_*(\Omega^2S^3;\mathbb{Z}/2)$ is polynomial on generators $y_i$ in degrees $2^{i+1}-1$ for $i\geq 0$, and the corresponding cohomology ring has generators $v_{ij}$ in degree $2^j|y_i|$ satisfying $v_{ij}^2=0$. These support many nontrivial Steenrod operations, which probably forces the map
$$ (\eta^*)^* \colon H^*(\Omega^2S^3;\mathbb{Z}/2) \to H^*(\Omega S^3;\mathbb{Z}/2)$$
to be mostly zero. All this story was worked out by Fred Cohen in his section of the book "Homology of iterated loop spaces".
It would also be good to understand $P_k$ when $k$ is odd. For this, we can use the following framework. Suppose we have a based space $B$ and a Hurewicz fibration $p\colon E\to B$ with fibre $F=p^{-1}\{*_B\}$. We can choose any point $x\in F$ and use it as the basepoint for $F$ and also for $E$; this then gives a fibration $\Omega_xF\to\Omega_xE\to\Omega B$ which gives information about the homotopy type of the component of $x$ in $F$. We can now take $B=\text{Map}_*(S^3,BU(2))$ and $E=\text{Map}_*(S^2,BU(2))$ with $p=\eta^*$; this gives $F=\text{Map}_*(\mathbb{C}P^2,BU(2))$. If we pick a point $x\in P_1\subset F$, we get a fibration as before, involving $\Omega_xF=\Omega_xP_1$. Here $E$ can be identified with $\Omega^2BU(2)=\Omega U(2)$, which has a group structure. This means that $\Omega_xE$ can be identified with the loop space at the usual basepoint, giving $\Omega_xE=\Omega^2U(2)=\Omega^2S^3$ (using $\Omega^2S^1=0$). Similarly, we have $\Omega B=\Omega^3U(2)=\Omega^3S^3$. Thus, $\Omega_xP_1$ is the fibre of a certain map $q\colon\Omega^2S^3\to\Omega^3S^3$. This is probably not exactly the same as $\eta^*$, because of the change of basepoint in $E$. It should be possible to work out the details, but I have not done so.
