Suggestion 1
If it's meant to be a monthly meeting why not make it a permanent "fourth Saturday of the month at blah blah timeslot" or whatever, and only the venue has to be arranged, then everyone knows the date and time beforehand and can plan around it.
Suggestion 2
If there are enough takers to make a reasonable turnup for both the 29th June and one of the dates in July would it be feasible to have 2 meetings then get into the swing of things with something definite for August i.e. my first suggestion?
I'm just being greedy - I want to pick brains ASAP!!!
Suggestion 3
I can see that making a permanent Saturday or Sunday (or whatever) may not suit some people (you can't suit all the people all the time) so if you sort something definite out maybe it could be odd months Jan, Mar, May etc could be Saturday and the even months Sunday.
Just a thought.
Cheers
Olly