Here's a version that might be useful to you.
– I would prefer "As I am now beginning my research project", but probably because my background is not American.
– perhaps "invite you along" is too informal, you decide. I'm trying to avoid writing "committee" yet again.
– your final sentence feels like two separate ideas clipped together by a semi-colon. I think the "guidance" part is a nice personal note and another small "thanks" sets it up.
Thank you for agreeing to serve on my committee. As I am beginning with my research project, Dr. King and I think it’s time for us to hold my first committee meeting. So I am writing this letter to invite you along and to arrange a suitable time. The available dates are: April 12th, 13th, 19th, 21st – please choose the one that is most convenient for you. Thank you again for your interest, your guidance is vital for my study. I look forward to your reply.