Don't forget that your tires aren't pulling that load; the engine and drivetrain are. (And the braking system has to BRAKE that load.)
That is an easy pull for the Trailblazer. Just read the many towing threads in here and be especially conscious of your tongue weight. Keep 10 to 15% of the total trailer weight on the tongue to reduce trailer sway. Also, tow in "3" not "D" and you'll do fine.
The best thing to do is to check the specs on the tire you like. It will tell you the maximum load it can carry. On my Michelin LTX M/S tires (which are probably the best tires you can get on our platforms, by the way) the max load is just under 2000 pounds. If you add up the total weight of the vehicle, the fluids, the cargo, every passenger PLUS the tongue weight of the trailer, you can see that you are still WAY under 4 times 2000 pounds.
If you consider only the back two tires, even when pulling a heavy trailer and with three hefty folks in the back seat, that is still lots of leeway.
With a fully loaded truck plus the tongue weight of the trailer, handling is not your priority; safe load-carrying ability is, so I would pump the rear tires up to very close to their max. rated psi as listed on the sidewall of the tire.
Hope this helps.
There is a lot more to tire pressures than most people realize. (Just don't get me started on nitrogen-filled tires, though!)