Bleichenbacher'06 attack can't be extended to the case at hand.
BB'06 attack with $e=3$ revolves on having the message representative $r$ (of format imposed by the 9 lines in the question's example) a cube of some integer $s$. Here the message representative $r$ is 128 bytes starting in 00 01 FF FF, with the $15$ leading bits at $0$ and several next ones at in, thus $r\lesssim2^{8\times128-15}=2^{1009}$, thus $s\lesssim2^{1009/3}$. More generally, for an $n$-bit RSA public modulus, the RSASSA-PKCS1-v1_5 message representative $r$ is $n'=8\,\lceil n/8\rceil-15$-bit, and just below $2^{n'}$, thus $s$ is just below $2^{n'/3}$. The BB'06 attacker's problem is finding a message and values XX such that $r=s^3$ for some $x$.
The derivative of $s^3$ w.r.t. $s$ is $3\,s^2$, therefore perfect cubes near $r$ are spaced about $3\,s^2=3\times2^{2\times1009/3}$. There are only $40$ bits in the free XX bytes that an adversary can choose, thus for a given hash only one chance in $2^{2\times1009/3-40}/3\approx2^{634.25}$ that suitable XX bytes exist if we don't use the fact there are many zeroes in the message representative.
That generalizes to: for an $n'$-bit message representative and arbitrary data imposed, we need at least in the order of $2\,n'/3$ free bits for the BB'06 attack to apply (and then their location can make the attack harder).
More rigorous argument: The last four lines in the example require that for some $x$ with $0\le x<2^8$ it holds $s^3\equiv\mathtt{0xAABBCCDD}\times2^{456}+x\pmod{2^{504}}$. If we exclude $x=0$, that leaves few values for $(x,s\bmod2^{504})$, that we can efficiently find and count: there are $224$, with haphazard values of $s\bmod2^{504}$, none of which less than $2^{1009/3}$ as required. Thus if there was a solution that's for $x=0$, thus the XX byte on the last line must be 00, and $s\equiv0\pmod2^{152}$. I'm quite confident that the last 32 free bits we have won't be enough to find a solution to the now haphazard-enough problem that we face (to be continued: I'm currently struggling at determining if there are solutions to $t^3\equiv\mathtt{0xAABBCCDD}\times2^{304}\pmod{2^{352}}$, where $s=t2^{152}\,$)
The question's example does not match the format described earlier in the question: there's no ASN.1 in the example, and (more importantly to the attempted rigorous argument) the example's AA BB CC DD is not in the format describedt.
