I think you will want to be careful coming into a new company. You may have the most experience in this area, but being new, you won't have critical knowledge about the company's culture. You mention that the current project lead has been in the company for a long time. Well, that person most likely has a good handle on how the company works. In a small company, it is important to understand this.
I know that you want very much to really shine in your new role, but make sure it isn't at the cost of another. I am not suggesting that you would undermine the project lead, but I would be careful about making suggestions that might make that person look bad. I would work with them and be diplomatic. Work to gently nudge the process in the right direction rather than attempting to ask for the reins. As a new employee, you want allies, but more than that, if you are really concerned with achieving the right solution, you need to be careful.
You may not need to be included in the "critical business design meetings". Why would you be there really unless to contradict a poor decision? And if this happens, you need to be extremely careful. A room full of people will be watching you and your project leader. When you are working with the project lead alone, you can carefully and tactfully discuss more appropriate approaches. This will help them to get better, let them know your level of expertise without being threatening and help to move the overall project forward in a positive way.
Over time, your skills will become depended on more and more. Work patiently and diplomatically toward a solution for everyone (you, the company AND the project lead). Be positive and helpful on as many levels as possible and I believe it will benefit your career and the overall solution in the best way.