Hooking up generator to house propane supply
Looks like the two regulators have the same output setting, and the BBQ supply is fed from a regulator with a high Btu/hr capacity. There are two more things you'll have to work out:

Does that 800,000 Btu/hr accommodate all the propane appliances that might operate concurrently?
With the added load of the generator drawing gas through the supply intended for BBQ, is the size of the system piping adequate to handle the flows required?

The second question can be more difficult to answer -- you'll need to find out the length, size, and material of the pipe in the gas system and then do some math. A web search for &quot;propane pipe sizing&quot; will turn up results that explain the methods -- for example, El Dorado County in California provides this handout which explains the &quot;longest length&quot; method of pipe sizing.