I'll offer my 60 active years of photography advice. The first thing is how big is the bank? Second is, "wild life" is a broad statement. Capturing a Bird sitting on a limb or a deer grazing motionless is one thing. Having a bird in flight with the sun directly overhead leaving the underside of the BIF in the shadows is something else ( you will be shooting the underside ). Are you looking for a Prime, or a Zoom? Are you expecting the Lens to auto lock the focus. or are you going to manually focus? You say you want 400mm or better.
A 400 mm plus prime will be thousands of dollars. A 400mm zoom with good glass will be in the 600 to 3000 range. You really would want a zoom lens with a focus limiter. A non focus limiting lens will Hunt more than it will focus lock. At least with a focus limiter the lens will have less time hunting and more time attempting a focus lock. This type of lens will be thousands of dollars.
I can tell you this. I would rather have one good long lens, and one good wide lens, that will do the job with little disappointment, then a bunch of inexpensive lenses that will be either a bitter disappointment or a bunch of dust collectors.
Lots of people want to be a wild life photographer, but have no idea how demanding action wildlife photography can be, especially to your pocketbook.
You can buy an inexpensive lens, pretend its expensive and hope to find people who will tell you what you want to hear here in this forum, or you may find someone that will be honest and tell you how poor a poor lens will perform. You will feel insulted, but as I said. save your money and buy a dedicated wildlife quality lens, not one with a caveat of being inexpensive or one that wont break the bank. Wildlife photography is an expensive discipline. A good enough lens gets old very fast.
Sorry for the long winded advice.. I give it as if I was giving it to my son or daughter
Good luck in your search.. BTW you are hardly alone in this quest.. many people come here every year with the same idea, no one comes back with the revelation that they bought a great inexpensive wildlife lens that didn't break the bank.
Bill