Im not quite sure, but i think its not that hard for US citizens to gain some kind of canadian residency. From doing that, you are then part of the commonwealth. I dunno if you could apply this way, but i would imagine it would be much easier to get a canadian visa/citezenship and apply that way then to actually find a job in britain, stay here for 4 years then apply.
I may be wrong, its just a idea? someone may be able to shine some light onto this.
Only 4 years? FFS you're hardly broken in. You ought to do a proper term of 22 years; it'll make a man out of ya.
If you're serious about wanting to join the British Army you might find it easier to do it through Canada which is a member of the Commonwealth. You might lose your US citizenship though.
Why not join the French Foreign Legion? If you pass selection they'll give you a 5 year contract no questions asked, a nom de guerre if desired and they'll teach you to speak soldier French complete with Froggy cuss-words.