Hello Brad!
I am going to tell you my opinion in my experience:
H3s in stock form and not used for heavy rock climbing, is more than enough with a set of 33s...35s makes sense if you are going to do a lot of rock crawling or very difficult trails, BUT you are going to suffer a significant drop in MPG and power (depends on your driving style, highway specs in your area (is not the same flat than highway hills) and tyre choose.
In my opinion, 90% of people just put 35s because esthetic, and not because improvement for offroad.
So here goes my understanding of perfect balance (super important concept that people forget) for a Hummer H3:
-Stay with 33s: It is the diameter designed for the vehicle from manufacturer, is the tyre designed for all the transmission components, brakes, axles, suspension, steering, etc, so is going to be the best balance between efficiency, strength and capability, but if you want to enhanced them, you can go a step further that is 305 70 R16 (33s but wider) I mounted them in my alpha, a set of Toyo Tyres Open Country MT and the result is awesome, on road and offroad (lot better) and seems like is a little bigger in diameter because of the tread of the MT.
-Spend some money on engine mods: This is by far the best way to enhance efficiency and power. You have the 3.7 i5, so you have the most efficient Hummer built, i recommend you this list:
_Airdoc Intake and AFE PRO Dry S filter
_Supermodulation Ported Throttle Body + supersparkz (still selling them on ebay)
_PCM of NC tune, adjusting after all the mods, for quicker shift points and quicker downshift under less throttle
With this, the H3 of my friend with BFs allterrain 33s is doing
8'9 seconds in the 0 to 60 mph (stock is 9'7 second with 32s), and
achieve a stable 20 MPG in highway trips at 65 mph (3.7 are far better than 3.5 in mpg), so
is aprox 270 hp at the flywheel
If you can go further and have the money I recommend you this next two mods for the engine:
_Orion Headers with a non fouler o2 in the second hole
_Dual efan setup (you can check my post here:
https://www.hummer4x4offroad.com/forum/showthread.php/14490-Guerrero-s-take-on-H3-cooling)
_Retune for this mods with PCM of Nc
With all of this, you will achieve a super balance fast (is not about fast, is about efficiency, more power equals to less mpg if well done) and capable truck. If you go the route of 33s bigger (305 70 R16) in MT you will see a super slightly less MPG, thats all
P.d: Of course I recommend you install some other important mods, crucial for h3s, like Tranny cooler bypassing stock tranny cooler in the stock radiator, Outfitter design kit for steering rack and front differential, and a good maintenance