Speaking from experience you BOTH need a type rating and line training for your MPL to become unrestricted. With just a type rating the MPL is restricted to your partner airline only. So you can't fly for other airlines. If your airline can't provide you with a TR then basically your licence is not complete and you're in trouble. An option in that case would be to revert back to the CPL/IR route. Up to PT1 (PPL) the training is the same. You'll sign some sort of contract with your partner airline so make sure this point is covered.