Say I'm feeding in few thousand bits data (INPUT AAAA) into both SHA256 & SHA3 256 engines at the same time. (Both engines using different hashing architecture) and hence it will generate different 256-bits of output, lets say SHA256 generate ABCD while SHA3-256 generate EFGH.
I'm curious about if we can try to find the alternative input (INPUT BBBB) later that can generate the same HASH output like above (SHA256 generate ABCD & SHA3 256 generate EFGH).
In short, this will be not be possible, even if we only use one secure hash function rather than two. You seem to be describing a sort of dual second-preimage attack where we need to find two inputs that clash over two separate hash functions.
A secure hash function will be resistant to such attacks. As such for either SHA2 & SHA3, it will not be possible to find another input that makes a desired output.
I believe that even SHA1 is only weak in terms of collision resistance. See here for further details.
