Typically, insurance will only allow one donor to do the testing at a time until they are ruled out. He may not necessarily be the better match. I know nothing about the HLA typing for livers.
The tests are to make sure that your husband is healthy enough to donate, both mentally and physically. A living liver donation is very major surgery and has significant risks that you will be informed about next week to make sure. At anytime, you can back out of donating and the center will simply say that you are not medically eligible.
Once a donor evaluation is complete, the coordinator and the surgeons will review all the information to determine if your husband is a good donor to his brother and if the surgery should take place. This typically takes place in a meeting with all the team members present. If he is approved, they will call with approval and discuss a time frame to get on the table.