my first brisket

If you really want to keep the Brinkman, I would consider getting a fire ring like they have in the bottom of the Weber Smokey Mountain and put it in the fire box. Use it to help keep a load of charcoal in the box. Maybe even consider using the Minion method for buring the charcoal. Depending how high you get the temps in your brinkman, you can also consider using foil (Texas crutch) to help finish the cooking process.

As you know, the brinkman has not insulation, leaks air into both the firebox and the cooking chamber besides other areas of concern. A good friend had a Hongo style cooker and went through all these problems you are facing now. In the end, I lent him one of my FEC 100s. He loved it and eventually bought one when one came up for sale here locally. While the FEC 100 may be more capacity than you need, the Cookshack electrics are an excellent way to go also.

Briskets (and pork butts) will plateau around 160-ish...meaning they take hours to rise out of that temp zone. They will come out but you need to be patient. Most briskets are done in the 195-200 range but the ultimate test is probing for tenderness over the 190 mark.

Google Ribdogs suggestions. I found some explanations and pic of the fire ring in another BBQ forum.

As for electric smokers, Cookshack has manufactured the Smokette for more than 50 years...they were designed to cook briskets. Load the meat, add a few chunks of wood and set temp. How easy is that?