I can't speak for the super light models but my standard Handi Hornet sure don't over heat and my Winchester M-70 sporter don't either. Like you I am a Woodchuck hunter but feel the Hornet would be ideal for moderate range colony varmints. In fact I think my 17hmr sporter heats up more than the Hornets, certainly not less.
Personally, with the compact nature and good handling of the Handi platform I don't think you need a Super Light in Hornet for walking. I have added a laminate stock (heavier than the wood) to mine and it's just fine. In fact it may be lighter than the pre war M-70 Hornet? A good medium is the pallet wood IMO. And it's much easier to tweak for accuracy if needed.
Last, I think you will be very pleased with the Handi Hornet. It just makes sense in break action rimed case. The barrels seem to be pretty good, likely just there old .223 barrels/twists chambered Hornet as they have a unusual 1/12 twist yet shoot the lighter stuff fine. I will tell you one thing I did do that I sure don't regret is, I ordered a butt plate from Brownells to replace that huge rubber recoil pad on the Hornet as the length of pull was to long for me to boot. Not sure you could line the holes up perfect for replacement with a plastic stock?
Last if you get one. You will find for factory ammo the Winchester 46 gr HP to be a shooter from the Handi. For reloads the standard 40 & 45 grain soft points, 40 gr V Max and some swear by the 35 V Max but I have not reloaded them yet. I keep it simple and just use Lil Gun and a Lee Classic Reloader.
It likes dirty and dry. I just run a dry bore snake through it now and then. I see posts of problems with other Handi calibers. Like the .223 I don't think you will see that with the Hornet. I will go out on a limb (from experiance) and say with factory Winchester 46 gr Hornet ammo you will get 3 shot moa @ 100 out of the box and maybe 5. BTW- thats a great woodchuck bullet! I punched one (reload same bullet) through upper center chest of a large woodchuck at 220 paces with a 4x scope from shooting sticks using the old Winchester M-70 that knocked him back deader than a doornail with out even a tail flag.
With a little accurizing or tweaked handloads you will likely get 5 shot moa @ 100 in low wind. I shoot from 100-300 at my rod & gun club but have a home 50 yard range that I test on often. My Handi now puts 5 reloads inside .5" at 50 like clock work, that includes the first from cold bore as I am a hunter and count it .
It will also do that with the Winchester 46 gr hp which is really a pretty darn good load and woodchuck bullet. Like to know what powder they use? Guys like speed and a few Hornet loads break 3000fps but that 46 gr Winchester load does about 2800 with a bullet thats 25% heavier than the 3000fps loads, hits hard and is accurate in many Hornets. I think after 70 years of Hornet Winchester had it figured out. I shot several Woodchucks at 175 -180 paces this summer with my Hornets.
I think you will really like it for what you want. Not sure if you even need to go to .221 Fireball or K Hornet as the accuracy is fine for the Hornets range and it will drop Woodchucks at 175- 200 yards with a solid upper body shot through Vidal's well. It's as cheap as buying rim mag ammo to reload for and Lil Gun powder really changed things. Personally, I like the 40,45 & 46 gr bullets over Lil Gun at about 2700 -2825 FPS for Woodchucks. They punch through often but just knock em deader than a doornail as a whole.
I think you will like it either way you go