Thank you for advice, but this is not the case.
The first attempt was to send externally signed TX with the response "16: ConnectInputs failed"
After the first attempt I did second one. It was to signe the TX in a hot wallet with imported private key that went well
And I have repeated this exercise with various unspent outputs more than 6 times and always with the same outputs.
So I came to conclusion that something special is done with the hex (that is hashed). If this is not the case, it should be something how the multichain test whether the sign, the message and TX are correct. But I have no idea how to solve this problem. Could you please give me some tips what should I try?