The electric grill is 1300 watts. It'll go through 50amp-hours (at 12 volts) to cook a couple of hamburgers for lunch in less than 20 minutes.
You can do this with LFP. The Bluetti plugged into the shore power connection as you illustrated would work. You may only get a couple meals out of it before the Bluetti runs out of juice. The coffee maker and microwave are high wattage loads also, but they only run for a few minutes. BBQ'ing is substantially longer.
The Bluettie unit you're describing is 2073 watt-hours. To put in perspective of 12volt, that's 160 amp-hours and can deliver 2,600 watts continuously. It'll definitely power the grill.
We have LFP on Channel Surfing and could run the electric grill, and still opted to leave the electric grill at home in the garage. For BBQ'ing, we use the propane stove with a griddle top. (LaZIna has a griddle pan she uses, or an aluminum rectangle, look like a cookie sheet, but it's heavier and has the ridges to grill food such as steak and such).