The length extension attack and security on length shortening of a hashed message by one byte MD5, SHA1, SHA2 are vulnerable to length extension attacks Wikipedia:Length extension attack.
Could is also be possible to generate $H(\text{message}[1..n-1])$ from $H(\text{message}[1..n])$ if I know the last byte?
It's essentially the same question as md5: is reverse length-extension attack possible?
Might it become possible, when I just want to shorten the message by just one byte in any of MD5, SHA1 or SHA2? Or could I at least reduce the computational cost of a brute force attack?
Could is also be possible to generate $H(\text{message}[1..n-1])$ from $H(\text{message}[1..n])$ if I know the last byte?

No, the length extension attacks are not working exactly like that. Let see how MD5 operates;
MD5 divides a message into 512-bit blocks to operate1 in Merkle–Damgård fashion way. Every message is padded. The messages are padded with 1 and following many zeroes so that the padded message size is multiple of 512 with the message length appended at the end represented in 64 bits. In a minimum way with always 1 is added the number of zeroes can be zero.
More formally, add bit 1 than add as many as required bit 0 until message length in bits $\equiv 448 \pmod{512}$ then add the message length in 64 bits. This also limits the file size that can be hashed with MD5.
So the $message[1..n-1]$ is calculated with $message[1..n-1] \mathbin\| padding$. After the padded message one can execute the length extension attack. The extended message with padding is;
$$\text{message}[1..n-1] \mathbin\| \text{padding} \mathbin\| \text{extension}\mathbin\|\text{padding}$$
To execute a length extension attack, one replaces the initial value of the target hash function with the hash. After this calculate the extended hash as usual hashing. Formally

let $\text{MD5}'(m,\text{MD5IV}) = \text{MD5}(m)$. I.e. $\text{MD5}'$ enables to control of the IV of the MD5.
let $h = MD5(m)$ of a message $m$.
then $h' = MD5'(m', h) = MD5(m\mathbin\|pad_1\mathbin\|m'\mathbin\|pad_2)$ where the $pad_1$ is the padding of $m$ when hashed with MD5 and $pad_2$ is the padding of $m\mathbin\|pad_1\mathbin\|m'$

The $h'$ is the length extended hash.

Might it become possible, when I just want to shorten the message by just one byte in any of MD5, SHA1 or SHA2? Or could I at least reduce the computational cost of a brute force attack?

After SHA3, there are variants of SHA2 like SHA512-256 that calculates a hash of output size 512 bits then truncates to 256. That almost eradicate the length extension attack's possibility. The SHA512-256 has a different initial value from SHA512-512 that separates the domains. In other words, they are different random oracles.
Of course, the pre-image resistance and secondary pre-image resistance and the collision resistance are weakened by $2^8$, $2^8$, and $\sqrt{2^8}$, respectively.

1The dividing is not specific to MD5 and diving size may change with each hash function like SHA512 uses 1024 bit block sizes