Some suggestions.
First ask your bank if it is possible to accept a cheque in US dollars. Some won't but some will, possibly for a fee.
Second ask your bank if there is an account you can open that would take a US dollar cheque. Lots of banks can set those up, again possibly for a fee. Try another bank in your country off they won't. This is your most likely solution.
Third ask the company if they can write a cheque in your home currency.
Fourth ask the company if they can write a money order instead if a cheque. They are similar enough that they might.
Fifth investigate what it takes to open an account in the US. If you are doing a lot of this it might be worth it. Your bank can probably help with this. They have partners and probably branches in the US.
Only then think about the process you are describing. It's going to be complicated so best to avoid it. You would need to get the cheque sent to you, endorsed, sent back, paid in, and then transferred.