Assume that is the correct manual for the OP's vehicle and is all
inclusive (there are no other sheets or modifier fine print)
So, 5.3L did NOT come from the factory with anything other than a 3.42
diff ratio(s)...unless it is used and the previous owner(s) changed
the diff ratio(s)...
Since this page lists GCWR, all the OP needs to do is load up as if
going camping (everything...people, pets, cargo, food, etc) in/on the TV

2009Sub 2009Suburban 2009SubGCWR 2009SuburbanGCWR
Then go out and weigh it, axle by axle preferred
Then do the simple math:
GCWR >= TV + trailer + stuff loaded
This includes the WD Hitch assembly
Since 5.3L, a half ton and regular half ton Sub's are in the mid 7K GVWRs
That would leave just under 7K for the trailer actual weight...not dry
For those who constantly argue not so...just take the GCWR listed and
subtract the listed 'curb' and it will be within the MTWR (there is a
+/- and that is for the hitch assembly/etc)