Looks like it is simple to implement however isn't there option to fetch Letter of Guarantee via api? I think it will be even better to give them signed message with the receiving bitcoin address just like when we are mixing with cryptomixer itself.
Yes. It is not listed on the website yet, but if you want to download the Letter of Guarantee you should use "Get Letter" method.
Get LetterThis method allows you to download the Letter of Guarantee.
POST /api/Letter
Host: cryptomixer.io
currencyId=BTC&address={INPUT_ADDRESS}
As the result you will get the LetterOfGuarantee.txt with the signed message inside. Simple.