One year and a half later I have the answer.
If your employer doesn't want to sponsor you with an LMIA, you still have other easier options to extend your stay, even without a bachelor's degree.
Tha catch is you need some money.
You can apply for a repeat participation of the Working Holiday with the help of a Recognized Organization - even if Canada does not normally allow that for your country of origin.
With a Recognized Organization, anybody from almost any country can apply for:
The requirements for the listed visas are the same as if you applied by yourself.
The only difference is that you'll have to pay a fee to the Recognized Organization and that can range from $500 CAD to $3000 CAD or more (it increases a bit every year).
Other (more difficult) options where higher education is not required are:
There are also many other immigration programs specific to each province that can help you.
All of the above does not apply to Quebec.