That would make it work better with the lift, but doesn't help my air tank clearance (in fact probably makes it worse).
The Toyota name is the LSPV, meaning load sensing proportioning valve.
The theory goes that as the suspension squats the rod pushes a plunger in the value up on the frame, which allows more brake fluid to reach the back brakes. If you follow the plumbing you'll see a hard line goes into the valve on the frame and the soft line from the axle goes up to it. On the axle the soft line splits to a pair of hard lines on the axle running to each wheel.
The factory sets the system to give just enough pressure not to lock up with an empty bed. There's a whole procedure in the service manuals on how to do it. This adjustment is why the bracket on the axle is threaded, you move the end of the rod up and down to set it.
The problem is when you lift the truck it appears like you've removed weight from the bed so the system actually reduces rear brake pressure further and effectively turns off the rear brakes. To solve this you can use the threaded adjuster, but it's usually too short to work. Some people bend the rod, which works if you can get it right but of course you never do, not to mention it's a permanent mod that doesn't allow you to life more or go back to stock.
So the bracket was invented I believe by Moses on his first mini truck. If you measure the amount it offsets it happens to be usually 2 or 3 inches. So it's a kludge, but it works and is simple.
As you have noted it all of this might solve the brake issue but doesn't solve your issue. Thing is the LSPV isn't the most sophisticated device in the first place. When new it works fine IMO and if you maintain it and take time to adjust things it can work after you lift. But usually it's only working marginally. So the alternative is to remove it altogether. However if you just bypass it you get way too much rear braking and will lock up the rear long before the front. On pavement this is a pain, on dirt it's a significant pain and on snow and ice this is suicidal.
Hod rodders have been faced with this problem forever. So they have simple proportioning valves they use on their custom brake systems. You do some searching to figure out what that mean, (I'd use "toyota mini truck brake proportioning valve" or something like that). You can get brake balance working with those just fine but the problem is there's no load-vs-no load change so you usually end up leaving some braking performance on the table when you're fully loaded. But it would let you eliminate the LSPV if you wanted.