1. We should settle the dispute through negotiations without resorting to legal proceeding.
我们应该通过仲裁解决争议而非法律途径。
还能这样说：
We should negotiate the dispute instead of resorting to legal proceeding.
We should not resort to legal proceeding but take the way to negotiate the dispute.
应用解析：
settle a dispute with sb. 同某人解决争端；
settle disputes between 调解……之间的争端；
dispute with sb. about 与某人争论某事

2. We prefer to resolve disputes by amicable, nonbonding conciliation between two parties.
我们愿意双方友好、互相谅解地解决争议。
还能这样说：
We would rather settle the disputes between two parties with an amicable and nonbonding conciliation.
We two parties settle the disputes in a friendly way with mutually understanding.
应用解析：
make a resolve to do sth. 决心做某事

3. If any dispute should arise over the inspection, we may submit it for arbitration.
如果是检验引起的任何争议，我们可接受仲裁。
还能这样说：
Any dispute caused by the inspection can resort to arbitration.
We may submit the dispute which arises over the inspection for arbitration.
应用解析：
arise from/out of 由……引起，由……而产生，从……中产生

4. If you are not prepared to compensate our loss, we suggest that case be submitted for arbitration.
若你们不赔偿我方损失，我们建议提交仲裁。
还能这样说：
We suggest submitting for arbitration if you don't pay for our loss.
We suggest that the case be submitted for arbitration in the event that you decide not compensates our loss.
应用解析：
suggest有意或无意地让对方察觉；
hint清楚的暗示，让对方明白自己的意向；
insinuate以巧妙的方法暗示不愉快或不能明说的事。