The problem is clearly that you do not have a clear agreement with the investor. All of the terms of the agreement (at least the most important ones) should be clearly lined out in a contract ASAP.
If you have someone on the team who does not contribute enough, then you need to see if you can resolve that with the person. If not, you need to find a way to remove that person from the team.
I saw a similar situation where an equity holder was part of the project to bring a product live. This was the first product so it was the foundation of the company. At any rate, the equity-holding member was a problem. Nice, but causing too many problems for others. They had to vote to remove him from the team. He could keep his shares but he would not be paid for his work going forward.
As you cam imagine, assuming you also hold equity, this is not ideal because the equity was given in the expectation of performance but you did not hire the team member and it sounds like you did not give up any of your equity. So, it is not really your issue to resolve. That said, if you can find a way to get the team member to perform, through sincere conversation, it would be better for everyone...especially if he has some skills you know you will need later.
Conversation must be the key here. You must find out why the member is not contributing more but you cannot force someone to be motivated, they have to want it. So, try to find out what will motivate them. I find that most people can answer the question "What motivates you?"