How to put a condition on a field for shipping methods in the Checkout

[GroupeCEA]

Hi,

We have two shipping methods:

1. Shipping by the customer's carrier that is marked 'free' and
2. Shipping by our company (there is a fee). 

The Shipping by the customer's carrier comes as default as it is the cheapest in view of the fact that Shopify displays the shipping methods from the cheapest to the most expensive.

However, we would like to have the Shipping by our company to come as default without using the BeSure Checkout rules app. How could we implement that change?

If that is not feasible, could this scenario be implemented:

1. A customer chooses Shipping by the customer's carrier as shipping method

2. The Checkout page requires the customer to indicate the name and the account number of his/her transporter.

How could this be implemented?

Here is the link of our store: https://pieces.groupecae.com/en 

Thank you for your support.