I appreciate all the responses! I had a feeling that this might be a bit of a unicorn and I would rather play it safe given the discussion I've seen here.
Fortunately I already have a PAL so I think I will give up the backyard/indoor shooting and simply go with something that would bring more power to the table.
It's not that much of a unicorn, really.
Despite fox-guy's assertions, the energy component does not come into play as a issue unless you are shooting over 500 fps.
You can shoot bowling balls at 499 fps, and it's not a firearm. An exaggerated example, but illustrative. Well, it is a firearm, sometimes, but not because of energy. It is, if you do stupid street thug stuff with it, and you will then be charged with it as if it was the same as Glock or Desert Eagle.
In reality, lots of guys have wandered into the rabbit hole of trying to make larger bore airguns and air pistols especially. There have been kits that allowed you to convert some Crosman models to 25 cal, 30, 9mm, and larger. They usually involve some serious modifications to the valving, to flow enough gas to be able to launch that heavy a projectile, plus a bunch of PITA work making your own ammo (in cases other than the .25 cal).
There are some big bore pellets available now, that have come available in recent years, but they are a mail order deal, not gonna find them at the local shop, for the most part.
So, that really means that you either live with the lower energy and flatter (somewhat) trajectory of a .177, or the higher energy of a .22 air gun. By our Laws, you are stuck with both being limited to the same velocity, and since gravity is constant, that gives both essentially the same trajectory. All else the same, that makes drag, the bigger difference. Even then, the larger mass of the .22 pellet... And so on...

So, if you want the energy of the larger pellet, you choose that. In either case, it makes the prospect of shooting game, into a accuracy fest. It's head shots, and even then, waiting until the angles are correct.
Well worth watching some of the British Airgun Hunting videos on youtube. They spend serious money on serious quality guns, so as to reliably place the pellet where they can use it to best effect. When you see a decent size rat just sorta quiver and drop in place from a well placed shot, you are watching the accuracy needed. They get it pounded in to them about humane kills. I could care less about whether a rat dies fast or slow. I just don't want one that runs away to ferment in the walls of the shop or barn. So, fast is the word of the day, as far as that goes. It requires the ability to place the shot exactly where needed. The kill zone is really, really small, dime size or less. If you cannot hit that reliably...well....
In the meanwhile, even if you cannot hunt with the same one you can shoot in your yard, you are still working up your hand/eye/muscle memory skills, and more trigger time is always a benefit to your shooting once you do go afield.
I have had some really great afternoons, sitting on the porch, shooting grasshoppers in my driveway, or off the lawn. Lining up hard candies, ice cubes, snow balls, etc., along the fence rail out to fairly far away, is a great way to estimate hold-over too. I much prefer reactive targets to groups on paper. YMMV.
Cheers
Trev