Sunday can still work, remember Sunday morning is usually church for many people and then people doing family dinners many time in the evenings. So on a Sunday it might be best to do say around 11 (have BBQ lunch) or maybe around 7-9 in the evening and still have BBQ but might not get as many to eat. Also something to think about, not sure of your area but in Texas it is hot right now. Best to make sure you have some shade for people, plenty of free bottle water for those visiting.
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